Theory

Differential equations

What differential equations and the Cauchy problem are, and the solution methods for first-order equations and for second-order equations with constant coefficients.

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Differential equations and the Cauchy problem
Local existence and uniqueness

Differential equations and the Cauchy problem

1. Differential equation, order, solution

A differential equation is an equation whose unknown is a function $y (x)$ and in which one or more derivatives of $y$ appear. The order of the equation is the highest order of differentiation present: $y^{'} = 2 y$ has order $1$ , $y^{''} + y = 0$ has order $2$ . A solution (or integral) on an interval $I$ is a function $y : I \to R$ , differentiable on $I$ as many times as the order requires, which, when substituted into the equation, turns it into an identity for every $x \in I$ . To verify that a function is a solution, one computes its derivatives and substitutes them into the equation.

Verifying a solution

$y = e^{3 x}$ is a solution of $y^{'} = 3 y$ on $R$ , since $y^{'} = 3 e^{3 x} = 3 y$ . The function $y = C e^{3 x}$ is also a solution for every constant $C \in R$ , because $y^{'} = 3 C e^{3 x} = 3 y$ .

2. General solution and particular solution

The general solution of a differential equation is the family of all its solutions; for the equations of order $n$ treated in this chapter it depends on $n$ arbitrary constants. A particular solution is the single solution obtained by assigning a value to the constants. The general solution of $y^{'} = 3 y$ is $y = C e^{3 x}$ with $C \in R$ ; for $C = 1$ one obtains the particular solution $y = e^{3 x}$ .

3. Cauchy problem

A Cauchy problem for a first-order equation is the system

{y^{'} = f (x, y) y (x_{0}) = y_{0}

formed by the equation and an initial condition. The initial condition selects, among the infinitely many solutions in the general solution, the one whose graph passes through the point $(x_{0}, y_{0})$ : one substitutes $x_{0}$ and $y_{0}$ into the general solution and solves for the constant.

Solving a Cauchy problem

For the problem $y^{'} = 3 y$ with $y (0) = 2$ , the general solution is $y = C e^{3 x}$ . Imposing the initial condition gives $y (0) = C = 2$ , so the solution is $y = 2 e^{3 x}$ .

y = 2 e^{3 x}

y = e^{3 x}

y = -e^{3 x}

Three solutions of

y^{'} = 3 y

y = 2 e^{3 x}

in color,

y = e^{3 x}

dashed,

y = - e^{3 x}

in red; the condition

y (0) = 2

selects the solution passing through

(0, 2)

Local existence and uniqueness

Let $f (x, y)$ be defined on an open rectangle of the plane containing the point $(x_{0}, y_{0})$ . If $f$ and the partial derivative $\frac{\partial f}{\partial y}$ are continuous on the rectangle, then the Cauchy problem $y^{'} = f (x, y)$ with $y (x_{0}) = y_{0}$ has one and only one solution, defined in a neighborhood of $x_{0}$ . The theorem guarantees that the problem is well posed before any computation. The adjective local indicates that the solution is guaranteed only near $x_{0}$ : the maximal interval on which it is defined can be smaller than the one on which $f$ is defined, as the exercises on the interval of existence show. The proof requires sequences of functions and is deferred to later courses.

Warning

Without the continuity of $\frac{\partial f}{\partial y}$ , uniqueness may fail. The Cauchy problem $y^{'} = 2 ∣ y ∣$ with $y (0) = 0$ has at least two solutions on $R$ : the zero function $y \equiv 0$ and the function $y = x ∣ x ∣$ , for which $y^{'} = 2∣ x ∣ = 2 ∣ y ∣$ . The function $f (x, y) = 2 ∣ y ∣$ is continuous, but its partial derivative with respect to $y$ is unbounded near $y = 0$ , so the hypothesis of the theorem is not satisfied.

y = x \left|x\right|

y = 0

Two solutions of the problem

y^{'} = 2 ∣ y ∣

y (0) = 0

y = x ∣ x ∣

in color and

y = 0

dashed, both passing through the origin

The theory continues with 6 more sections (separable equations, first order linear equations, bernoulli equations…), plus all worked examples, the solutions and the PDF.

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