Equations and Solution Spaces

5.1Function Spaces and Solution Sets

Start with the spring. We already know two of its solutions, cos⁡𝑡 and sin⁡𝑡. Suppose 𝑢 and 𝑣 are any two solutions of 𝑥″ = −𝑥, and add them. Differentiation respects sums, so

The sum is another solution! The same calculation works for a multiple 𝑐𝑢, whose second derivative is 𝑐𝑢″ = −𝑐𝑢. So starting from just cos⁡𝑡 and sin⁡𝑡, scaling and adding produces every function 𝐴cos⁡𝑡 +𝐵sin⁡𝑡 in the family we wrote down. We can build new spring solutions out of old ones as freely as we like.

Now try the same thing with 𝑦′ =𝑦2, whose solutions taught us in chapter 3 that a perfectly smooth rule can still produce finite-time blowup. If 𝑢 and 𝑣 are two solutions, then

But for 𝑢 +𝑣 to be another solution, the equation would require

The extra term 2𝑢𝑣 vanishes only where one of the two solutions does. Differentiation respects addition, but squaring does not: the sum of two solutions follows the wrong rule. Scaling fails the same way, since (𝑐𝑢)′ =𝑐𝑢2 while the equation demands (𝑐𝑢)2 =𝑐2𝑢2.

Look at what we just did. We added two whole functions, multiplied a function by a number, and asked whether the result still solved the equation. That is the arithmetic of vectors. A list of three numbers counts as a vector because we can add lists and multiply them by scalars, and linear algebra applies the same idea to matrices, polynomials, and functions. It may have been a while since you thought of a function as a vector, but that is exactly the viewpoint we need for studying a whole collection of solutions at once.

A solution is always a solution on an interval. On an interval 𝐼, we need a function with enough derivatives to make every expression in the equation meaningful, satisfying the equation at every time in 𝐼. The interval matters. The function 𝑦(𝑡) =1/(1 −𝑡) solves 𝑦′ =𝑦2 on ( −∞,1), but cannot provide a solution through 𝑡 =1. A formula and the interval on which it satisfies the equation belong together.

Fix an open interval 𝐼. If 𝑓 and 𝑔 are real-valued functions on 𝐼, their sum and a scalar multiple are defined by

These are pointwise operations: perform the arithmetic separately at each input to obtain a new function. The familiar algebraic rules hold because they hold at every input. For instance, 𝑓 +𝑔 =𝑔 +𝑓 because 𝑓(𝑡) +𝑔(𝑡) =𝑔(𝑡) +𝑓(𝑡) for every 𝑡. The zero vector is the function which is zero everywhere, and the additive inverse of 𝑓 is the function −𝑓. Thus real-valued functions on a fixed interval form a vector space.

For differential equations, we need functions we can differentiate. Write 𝐶𝑘(𝐼) for the space of real-valued functions on 𝐼 with continuous derivatives through order 𝑘. In particular, 𝐶1(𝐼) consists of continuously differentiable functions, while 𝐶2(𝐼) requires two continuous derivatives. These are vector spaces too: adding or scaling functions preserves the required differentiability. We use 𝐶(𝐼) for the space of continuous functions. The linear algebra appendix reviews these spaces alongside the familiar spaces of numerical vectors.

These spaces are enormous! Even 𝐶𝑘(𝐼) for a fixed 𝑘 contains all polynomials. The functions 1,𝑡,𝑡2,… are linearly independent: a finite linear combination that vanishes throughout an interval is a zero polynomial, so all its coefficients vanish. Consequently, 𝐶𝑘(𝐼) contains arbitrarily large linearly independent collections. It is infinite-dimensional; no finite list of functions can form a basis for the whole space.

We will not need to find a basis for such a space. Its role is to hold the functions among which we are looking for solutions. For example, the solutions of 𝑦′ =𝑦 on 𝐼 form the subset

This notation separates the two requirements. Membership in 𝐶1(𝐼) makes differentiation available; the condition after the colon asks that the derivative agree with the function. We call S the solution set, and 𝐶1(𝐼) its ambient space. For a second-order equation we can work in 𝐶2(𝐼) instead. In each case, the equation selects a subset of a much larger space of functions.

Systems fit into the same picture. A solution of the system 𝑓′ =𝑔, 𝑔′ = −𝑓 from the introduction is a pair of functions, and neither is an answer on its own: the derivative of the first must equal the second, and the derivative of the second must equal the negative of the first. We can regard the pair as a single vector-valued function,

At each time, 𝑋(𝑡) is a vector in ℝ2. The whole function 𝑋 records how that vector changes with time. Its derivative is computed component by component, so the system can be written

This is still the same pair of equations. Collecting the functions into a vector lets us speak of one solution while keeping track of all its components.

More generally, a first-order system for 𝑁 unknown functions, with the equations solved for their derivatives, has the form

Each rate may depend on every current value. Collecting the unknowns into 𝑋 =(𝑥1,…,𝑥𝑁) and the known rules into 𝐹 =(𝐹1,…,𝐹𝑁) gives the compact expression 𝑋′ =𝐹(𝑡,𝑋). The functions 𝐹𝑖 may contain products, squares, or other nonlinear expressions, as they did in our population models. They may also depend explicitly on time. Vector notation accommodates all of these possibilities.

For systems, use 𝐶𝑘(𝐼,ℝ𝑁), the space of vector-valued functions whose components belong to 𝐶𝑘(𝐼). Addition, scaling, and differentiation act component by component. Notice that 𝑁 counts the components of the value 𝑋(𝑡) at one time; it is not the dimension of this function space. Even one component can be any of infinitely many linearly independent functions.

5.1.1Parameters for a Family of Functions

How large is the subset selected by an equation? We can begin with the families we already know. On ℝ, every solution of 𝑦′ =𝑦 has the form 𝑦𝐶(𝑡) =𝐶𝑒𝑡. The assignment

takes a number and returns an entire function. Changing 𝐶 moves through the solution set. Every solution occurs, and two different values of 𝐶 give different solutions: just evaluate at 𝑡 =0 to recover 𝑦𝐶(0) =𝐶. Thus one real number gives a coordinate for each member of this family.

Geometrically, these functions form a line through zero in function space. They are precisely the scalar multiples of the nonzero vector 𝑒𝑡. This line looks quite different from the collection of exponential graphs we drew earlier. Each entire graph represents just one point of the line! Moving along the graph means varying 𝑡 within one function. Moving along the line in function space means varying 𝐶 to choose a different function.

The spring gives us a family with two parameters:

Its parameters can be recovered from 𝑦𝐴,𝐵(0) =𝐴 and 𝑦′𝐴,𝐵(0) =𝐵, so different pairs give different functions. These functions form the plane spanned by the vectors cos⁡𝑡 and sin⁡𝑡 in 𝐶2(ℝ). Their independence is easy to check: if 𝐴cos⁡𝑡 +𝐵sin⁡𝑡 is the zero function, its value and derivative at zero force 𝐴 =𝐵 =0. Choosing a point (𝐴,𝐵) in the parameter plane therefore chooses one point in this plane of functions.

This is how a differential equation can leave us with finitely many choices even though we began in an infinite-dimensional space. We have already verified that this whole plane consists of spring solutions. Proving that it contains every spring solution is a further claim; the existence-and-uniqueness theory will help us settle claims of this kind without guessing what other functions might work.

5.1.2Linear Differential Equations

A subset of a vector space is a vector subspace if it contains zero and is closed under linear combinations. Thus, for a solution set to be a vector space, the zero function must solve the equation, and whenever 𝑢 and 𝑣 solve it, 𝑎𝑢 +𝑏𝑣 must solve it for all real numbers 𝑎,𝑏. The exponential line and the plane of spring solutions have this property. Can we recognize it from the equations themselves?

Differentiation already respects the arithmetic we need:

In the language of linear algebra, differentiation is a linear map from 𝐶1(𝐼) to 𝐶(𝐼). Its input is a function, and its output is another function. More generally, a map 𝐿 between vector spaces is linear when 𝐿(𝑎𝑢 +𝑏𝑣) =𝑎𝐿(𝑢) +𝑏𝐿(𝑣). We often call a map acting on functions an operator.

Consider the operator

It takes a function, differentiates it twice, and adds the original function. Both operations respect linear combinations, so

Our spring equation 𝑦″ = −𝑦 is exactly the equation 𝐿[𝑦] =0. Its solutions are the functions the operator sends to zero: the kernel of 𝐿. Recall why a kernel is a subspace. It contains zero, and if 𝐿[𝑢] =𝐿[𝑣] =0, then 𝐿[𝑎𝑢 +𝑏𝑣] =𝑎 ⋅0 +𝑏 ⋅0 =0. We have proved that the entire solution set is a vector space without first finding all its members!

This reasoning works for much more than the spring. Multiplication by a fixed function of time is also linear: for any known coefficient 𝑝(𝑡), 𝑝(𝑡)(𝑎𝑢(𝑡) +𝑏𝑣(𝑡)) =𝑎 𝑝(𝑡)𝑢(𝑡) +𝑏 𝑝(𝑡)𝑣(𝑡). Consequently, an operator of the form

is linear. With continuous coefficients, it maps 𝐶𝑛(𝐼) to 𝐶(𝐼). The coefficients can change with time; what matters is that they are fixed functions, independent of the unknown 𝑦.

The most familiar case is a scalar equation of order 𝑛,

whose operator is the one we just built. For example, 𝑦′ =𝑦 is the homogeneous linear equation 𝑦′ −𝑦 =0, and 𝑦′ =𝑡𝑦 is homogeneous and linear too: the coefficient 𝑡 varies, but the dependence on the unknown is still linear. Squaring 𝑦 or 𝑦′ would destroy the linearity of the corresponding operator.

For a system, consider

Its operator sends the pair 𝑋 =(𝑥,𝑦) to 𝐿[𝑋] =(𝑥′ −2𝑦,𝑦′ +3𝑥). This is a linear map from 𝐶1(𝐼,ℝ2) to 𝐶(𝐼,ℝ2), and once again the solution set is its kernel. To give this space some concrete members, let 𝜔 =√6. Direct differentiation verifies the family

Here 𝑥(0) =𝐴 and 𝑦(0) =𝐵. In checking the second equation, the identity 𝜔2 =6 makes the coefficients agree. Adding two of these vector-valued solutions adds their parameters; multiplying a solution by a scalar multiplies both parameters by that scalar.

The same kernel argument applies to all these equations, regardless of their order or number of components.

The proof is the kernel calculation above. The word linear describes this arithmetic of functions, not the shapes of their graphs. Our solutions can grow exponentially or oscillate while still belonging to a vector space.

5.1.3Affine Solution Sets

What about a linear equation with a nonzero right-hand side? We have been solving these since antidifferentiation. For example, 𝑦′ =2𝑡 has the full solution family

This is a line in function space too, but it does not pass through zero: the zero function does not have derivative 2𝑡. Adding two solutions gives 2𝑡2 +𝐶1 +𝐶2, whose derivative is 4𝑡, so the sum leaves the solution set. Yet subtracting two solutions gives the constant 𝐶1 −𝐶2. The direction of the line is the space of constant functions, which solves the associated homogeneous equation 𝑦′ =0.

Recall that an affine space inside a vector space is a translate 𝑝 +𝑉 ={𝑝 +𝑣 :𝑣 ∈𝑉} of a vector subspace 𝑉. The point 𝑝 locates the translate, while 𝑉 describes its directions. Our family 𝑡2 +𝐶 is the translate of the constant functions by the particular function 𝑡2. The constants of integration were describing an affine space all along.

The same relationship holds for every solvable linear equation. Suppose 𝐿[𝑢] =𝑔, and we have found one particular solution 𝑢𝑝. If 𝑢 is any other solution, linearity gives

Thus 𝑢 −𝑢𝑝 belongs to ker⁡𝐿. Conversely, adding any homogeneous solution ℎ ∈ker⁡𝐿 to 𝑢𝑝 gives 𝐿[𝑢𝑝 +ℎ] =𝑔 +0 =𝑔. These two calculations establish the whole solution set:

When 𝑔 =0, we can choose 𝑢𝑝 =0 and recover the vector space ker⁡𝐿. When 𝑔 is not the zero function, zero is not a solution, so the affine solution set is not a vector subspace. In either case, this description assumes that a particular solution exists; it does not yet prove existence for arbitrary equations or initial data.

5.1.4Nonlinear Solution Sets

For a linear equation, we now know which arithmetic preserves solutions: linear combinations when the equation is homogeneous, and a particular solution plus a homogeneous one when it is forced. The calculation for 𝑦′ =𝑦2 showed that addition and scaling need not preserve solutions of a nonlinear equation. The logistic equation 𝑦′ =𝑦(1 −𝑦) has the same difficulty. For general nonlinear equations, we have no superposition principle: knowing several solutions does not, by itself, give a way to construct the solution from another initial state.

For a second-order comparison, place the spring 𝑥″ = −𝑥 beside

Here 𝑥(𝑡) =𝑡 and 𝑥(𝑡) = −𝑡 are two solutions: each has second derivative zero and first derivative with square one. Their sum is the zero function, whose derivatives both vanish, so substituting it would require 0 =1. Doubling fails too: 𝑥 =2𝑡 has 𝑥″ =0, but 1 −(𝑥′)2 = −3. A nonlinear dependence on a derivative breaks the arithmetic just as a nonlinear dependence on the function does.

Yet this equation has plenty of solutions. Here is a whole family of them:

We do not need to discover this formula to make use of it; we can check it. Write ℎ(𝑡) =cosh⁡𝑡 +𝑏sinh⁡𝑡, so that ℎ″ =ℎ. Wherever ℎ >0, differentiating the proposed solution gives

The formula works. And since ℎ(0) =1 and ℎ′(0) =𝑏, its parameters are exactly the initial data:

We can choose initial position and initial velocity independently, just as we could for the spring. But they enter this family differently. Changing 𝑎 translates the graph vertically, while changing 𝑏 changes the function inside the logarithm. In fact, adding any constant to a solution preserves this equation, since its derivatives do not change. So we can obtain some new solutions from a known one. What we lack is the freedom to combine solutions by arbitrary addition and scaling. We still have two independent initial-data parameters, even though they do not enter as the coefficients of two fixed solutions.

For any real 𝑎,𝑏, the formula is defined on an interval around zero, because ℎ(0) =1. If |𝑏| ≤1, it is defined for all real 𝑡:

In particular, 𝑏 =1 gives 𝑥 =𝑎 +𝑡, and 𝑏 = −1 gives 𝑥 =𝑎 −𝑡, recovering our earlier solutions. For |𝑏| >1, ℎ vanishes at a finite time on one side of zero; we use the interval containing zero on which it is positive. The parameters can affect how long a solution exists, as well as the shape of its graph.

The spring family and the logarithmic family each use two numbers, the initial position and velocity, to pick out one member. Their different arithmetic does not change that count. So far, though, every count like this has come from a formula. How many numbers pick out a solution when there is no formula to read them from? Chapter 3 answered this for a single first-order equation 𝑦′ =𝐹(𝑡,𝑦): under the theorem's hypotheses, one initial value determines a unique local solution, even when we cannot find a formula. We would like the same kind of answer for every equation we have met.

5.2A Common Formulation: First-Order Systems

We already found a connection between higher-order equations and systems when modeling the spring. Its equation 𝑥″ = −𝑥 tells us the acceleration from the position, but position alone does not determine the motion. Two masses can pass through the same position with different velocities. To describe how either one moves, we need to keep track of both quantities.

Give velocity its own name, 𝑣 =𝑥′. We can now ask for the first derivative of each quantity we are tracking. The derivative of position is velocity, and the derivative of velocity is the acceleration supplied by the equation:

We have replaced one second-order equation with two first-order equations. The new variable records information the original problem already needed. The initial conditions 𝑥(𝑡0) =𝑥0 and 𝑥′(𝑡0) =𝑣0 become the single vector condition 𝑋(𝑡0) =(𝑥0,𝑣0) for 𝑋 =(𝑥,𝑣).

Nothing about this construction required the spring equation to be linear. Apply it to our nonlinear example 𝑥″ =1 −(𝑥′)2. With the same choice 𝑣 =𝑥′, we obtain

The second equation now involves only 𝑣, so we can solve it by separation and then integrate 𝑥′ =𝑣. The solutions 𝑥 =𝑡 and 𝑥 = −𝑡 correspond to the system solutions (𝑥,𝑣) =(𝑡,1) and (𝑥,𝑣) =( −𝑡, −1). In the system, we can see why those motions work: the velocities 1 and −1 make 𝑣′ =0 and remain constant.

More generally, consider a second-order equation

The same substitution produces 𝑥′ =𝑣, 𝑣′ =𝐺(𝑡,𝑥,𝑣). We should check that these descriptions really have the same solutions. If 𝑥 solves the scalar equation on an interval 𝐼, define 𝑣 =𝑥′. Then 𝑥′ =𝑣 by definition, and 𝑣′ =𝑥″ =𝐺(𝑡,𝑥,𝑣) by the original equation. Thus (𝑥,𝑥′) solves the system on 𝐼.

Conversely, suppose a pair (𝑥,𝑣) solves the system. Its first equation forces 𝑣 =𝑥′. Differentiating gives 𝑥″ =𝑣′ =𝐺(𝑡,𝑥,𝑣) =𝐺(𝑡,𝑥,𝑥′), so its first component solves the original scalar equation. These constructions undo each other. We have a one-to-one correspondence between the two solution sets, including the solutions with any specified initial position and velocity. No existence or uniqueness theorem was needed to establish this correspondence; it follows directly from the equations.

5.2.1Recording the Derivatives

What happens for a third-order equation? Recording 𝑦 and 𝑦′ is no longer enough to write down their rates: the derivative of 𝑦′ is 𝑦″, which the equation has not yet determined from those two values. So record 𝑦″ too. Its derivative is 𝑦‴, and that is what the third-order equation supplies. The construction stops when we have recorded all derivatives below the highest one.

For an equation of order 𝑛 ≥2 written as

introduce the coordinates

Differentiating each coordinate gives the first-order system

The first 𝑛 −1 equations record the meaning of our new coordinates. The last carries the original differential equation. Thus the state is 𝑋 =(𝑦,𝑦′,…,𝑦(𝑛−1)), and prescribing its value at 𝑡0 specifies exactly the usual 𝑛 initial values for the scalar equation.

The equivalence argument works just as it did for order two. A scalar solution produces a system solution by collecting its derivatives. Starting with a system solution, the first equation gives 𝑥2 =𝑥′1, the next gives 𝑥3 =𝑥″1, and continuing gives 𝑥𝑛 =𝑥(𝑛−1)1. The last equation then says that 𝑦 =𝑥1 solves the original equation. In particular, the coordinates of a system solution cannot be chosen as unrelated functions: the derivative relations tie them together.

For linear equations, this conversion also preserves linearity. If 𝑎𝑛(𝑡) is nonzero, divide 𝑎𝑛𝑦(𝑛) +⋯ +𝑎1𝑦′ +𝑎0𝑦 =𝑔 by 𝑎𝑛 to obtain the last system equation

Together with the derivative chain, this is a linear system, homogeneous when 𝑔 =0. The correspondence 𝑦 ↦(𝑦,𝑦′,…,𝑦(𝑛−1)) itself respects addition and scaling. Our two descriptions therefore agree about the linear or affine structure of their solution sets too.

Every first-order linear system has this shape. Collecting the coefficients into a matrix, it reads

where 𝐴(𝑡) is an 𝑁 ×𝑁 matrix of known functions and 𝑔(𝑡) is a known vector-valued function. In the language of definition 5.1, its operator is 𝐿[𝑋] =𝑋′ −𝐴(𝑡)𝑋 and its forcing is 𝑔. The converted scalar equation is one example; the system 𝑥′ =2𝑦, 𝑦′ = −3𝑥 from the previous section is another, with

This is the form the next chapter starts from.

5.2.2Coupled Equations and the State

The same construction applies when several equations involve higher derivatives. Suppose 𝑞 =(𝑞1,…,𝑞𝑚) is a vector of unknown functions satisfying

Record the vector 𝑣 =𝑞′ along with 𝑞. We obtain 𝑞′ =𝑣 and 𝑣′ =𝐺(𝑡,𝑞,𝑣), a first-order system with 2𝑚 scalar coordinates. Each component contributes its value and its first derivative to the state. Equations with different orders work the same way: for each unknown, record its derivatives up to one below the order of the equation that determines its highest derivative. If those orders are 𝑛1,…,𝑛𝑚, the resulting state has 𝑁 =𝑛1 +⋯ +𝑛𝑚 coordinates. This assumes the original system has been solved for those highest derivatives in terms of time and the recorded quantities.

Equations solved for their highest derivatives in this way are said to be in normal form, and the conversion applies to them. A scalar linear equation can be put in normal form wherever its leading coefficient 𝑎𝑛(𝑡) is nonzero; at a zero of 𝑎𝑛 we cannot divide.

Within this scope, all the equations now have the same shape:

Here 𝑋 is exactly the state of definition 1.3: the current values and derivatives we must record as initial data. The differential equation supplies their rates of change. The starting time 𝑡0 is also part of the initial value problem: when 𝐹 depends explicitly on time, the same state prescribed at different times can have different rates.

Could we reduce everything to one scalar first-order equation instead? Even the spring shows what would be lost. At a fixed time and position, we may choose different velocities, while a scalar equation 𝑥′ =𝑓(𝑡,𝑥) assigns just one velocity. Recording a vector allows us to retain both initial choices. We have reduced the order by keeping more coordinates; we have not removed the information in those coordinates.

This leaves us free to use whichever description is convenient. The scalar equation 𝑥″ = −𝑥 makes the relationship between acceleration and position immediate, while its system makes the full state explicit. A theorem about first-order systems can now apply to both descriptions. The next question is whether specifying 𝑋0 actually determines a solution: does a history begin at every initial state, and can more than one history begin there?

5.3Existence, Uniqueness, and Initial Data

For one first-order equation, we already have a theorem that answers this question. Continuity of the rate function and its derivative with respect to the unknown gives local existence and uniqueness. For a system, the corresponding hypotheses ask us to check how each component of the rate depends on each component of the state.

Write 𝐹 =(𝐹1,…,𝐹𝑁) and 𝑋 =(𝑥1,…,𝑥𝑁). The partial derivative 𝜕𝐹𝑖/𝜕𝑥𝑗 measures how the 𝑖th rate changes as we vary the 𝑗th state coordinate, holding time and the other coordinates fixed. There are 𝑁2 such derivatives. The theorem has the same form in every dimension.

We will use the system theorem without proving it. Our scalar theorem is its 𝑁 =1 case; the conversion of higher-order equations tells us how to apply the same result far beyond that case. The hypotheses are sufficient conditions, not a claim that every equation failing this test must fail to have a unique solution.

For example, our nonlinear second-order equation became

Here 𝐹1(𝑥,𝑣) =𝑣 and 𝐹2(𝑥,𝑣) =1 −𝑣2, with state derivatives

Everything is continuous everywhere. Thus any choice of 𝑥(𝑡0) and 𝑣(𝑡0) gives a unique local system solution. By the correspondence we proved, any choice of 𝑥(𝑡0) and 𝑥′(𝑡0) gives a unique local solution of the second-order equation. We have answered the question for every initial position and velocity without solving the equation for any of them.

The same reasoning applies to an order-𝑛 equation in normal form. If 𝐺 and its partial derivatives with respect to 𝑦,𝑦′,…,𝑦(𝑛−1) are continuous near the prescribed data, then the converted system satisfies the theorem's hypotheses. Each derivative-chain equation has a coordinate as its right-hand side; only the last equation requires us to check 𝐺. The 𝑛 prescribed initial values therefore determine a unique local scalar solution. Coupled higher-order equations are handled by the same check on their combined state.

Uniqueness also settles a question we left open about the spring. We verified that 𝑥(𝑡) =𝐴cos⁡𝑡 +𝐵sin⁡𝑡 solves 𝑥″ = −𝑥. Given any solution on an interval containing zero, take 𝐴 =𝑥(0) and 𝐵 =𝑥′(0). The displayed formula and the given solution then have the same initial state, so uniqueness forces them to agree near zero. They agree throughout their common interval: if agreement could stop at some time inside that interval, their states there would agree by continuity, and local uniqueness at that time would extend the agreement. This is the same continuation argument we used for scalar first-order equations. There are no missing spring solutions! The plane of functions we found is the whole solution set on ℝ, and its restrictions give the solutions on smaller intervals containing zero.

Existence is still a local guarantee. Even when the hypotheses hold everywhere, a solution need not exist for all time. We already know this from 𝑦′ =𝑦2: the solution with 𝑦(0) =𝑎 ≠0 is 𝑦(𝑡) =𝑎/(1 −𝑎𝑡) on its interval through zero, and it blows up at 𝑡 =1/𝑎. Different initial values can therefore give different intervals of existence. The theorem guarantees a history near the initial time, not that we can follow it indefinitely. In fact, only the zero solution of 𝑦′ =𝑦2 exists on all of ℝ.

Uniqueness matters just as much. The functions 𝑦 =0 and 𝑦 =𝑡3 from Chapter 3 solve 𝑦′ =3𝑦2/3 with the same value at zero. For that equation, an initial value cannot distinguish the two histories. When we use initial data to identify a solution, we are using a theorem about the equation, not merely counting the constants in a formula.

5.3.1Initial Data as Parameters

How can finitely many numbers specify an entire function? A general function can take values independently at many different times. A solution of a differential equation has much less freedom: its values must fit together according to the equation. Existence and uniqueness tell us just how much freedom remains.

Fix an initial time 𝑡0. Under the hypotheses of our theorem, each admissible initial state 𝜉 determines a unique local solution 𝑋𝜉, with 𝑋𝜉(𝑡0) =𝜉. Existence supplies a solution near 𝑡0. Uniqueness means that any two solutions with those data agree throughout their common interval containing 𝑡0. Going in the other direction is easy: given a solution defined near 𝑡0, evaluate it there to recover the state that selected it.

The initial state therefore works as a set of coordinates for local solutions. For an equation whose state has 𝑁 coordinates, 𝑁 numbers recorded at a single time determine the solution near that time. This is remarkable information to have without a solution formula: we know how many initial values specify a solution before we have found a single one.

This answers the question we asked of our formulas. The count comes from the equation, through its state, not from how many constants happen to appear in a formula. We could write the solutions of 𝑦′ =𝑦 as (𝐴 +𝐵)𝑒𝑡, but only the sum 𝐴 +𝐵 matters. The state of 𝑦′ =𝑦 has one coordinate, and one number is what it takes.

The local qualification matters. Different initial states can give solutions on different intervals, as the blowup of 𝑦′ =𝑦2 showed. Restricting a solution to a smaller interval containing 𝑡0 also preserves its initial data. The data determine how the solution behaves where it is defined; they do not prescribe which interval we write it on.

For a higher-order scalar equation, remember what these coordinates mean. We first regard its solution as the vector-valued solution 𝑋 =(𝑦,𝑦′,…,𝑦(𝑛−1)). Evaluation then records

So the order-𝑛 scalar equation has a locally 𝑛-parameter solution family under the hypotheses above. The system formulation explains why derivative values count as parameters in exactly the same way as the component values of a first-order system.

Return now to the nonlinear family we checked earlier:

We verified that each function solves the equation wherever the logarithm's argument is positive. Have we found all the solutions near zero, or could there be others that our formula missed? Take any solution 𝑥 on an interval containing zero and record 𝑎 =𝑥(0), 𝑏 =𝑥′(0). The function 𝑥𝑎,𝑏 exists near zero and has exactly those initial data. Uniqueness forces 𝑥 =𝑥𝑎,𝑏 near zero, and the continuation argument extends their agreement throughout their common interval. There are no missing local solutions. We did not need a systematic method for solving the equation to establish that our verified family was complete!

We began this chapter by asking two questions about the collection of all solutions of an equation. What kind of set is it? Homogeneous linear equations give vector spaces, and fixed forcing gives translates of those spaces whenever a particular solution exists. Nonlinear equations generally have neither structure, so the same reconstruction by linear algebra is not guaranteed. How many initial values specify a local solution? As many as there are coordinates in the state, for every equation our theorem covers.

The spring and 𝑥″ =1 −(𝑥′)2 show that these answers are separate facts. Both need two numbers. For the spring, those numbers select the combination 𝑥(0)cos⁡𝑡 +𝑥′(0)sin⁡𝑡; for the other equation, they enter the logarithmic formula in a different way. Knowing how many numbers we need does not tell us what arithmetic the solutions allow.

For homogeneous linear equations, these two kinds of information fit together remarkably well. Evaluating at 𝑡0 respects addition and scaling, and those operations keep us inside the solution set. In the next chapter we put the two together: the initial state becomes a set of linear coordinates for whole solutions, which will tell us the dimension of the solution space, give us bases of solutions, and describe evolution in time by a matrix.