Differential equations and their solutions

What a differential equation is, its order, checking that a function is a solution, general and particular solutions, families of solution curves, and using initial conditions to find constants.

What is a differential equation?

A differential equation (DE) is an equation involving a derivative, such as $\dfrac{dy}{dx} = 2x$ or $\dfrac{d^2y}{dx^2} + y = 0$. Its order is the order of the highest derivative in it.

A solution is a function that makes the equation true. To check one, substitute it and its derivatives into both sides.

Example 1●●

Show that $y = \sin x$ is a solution of $\dfrac{d^2y}{dx^2} + y = 0$.

$\dfrac{dy}{dx} = \cos x$ and $\dfrac{d^2y}{dx^2} = -\sin x$.

So $\dfrac{d^2y}{dx^2} + y = -\sin x + \sin x = 0$, as required.

General and particular solutions

The general solution contains arbitrary constants and describes a whole family of curves. A first-order DE usually needs one constant, and a second-order DE two. An initial condition, such as $y = 3$ when $x = 0$, picks out one particular solution.

A family of solutions

Every curve y = x² + C solves dy/dx = 2x. The initial condition chooses which one.

Try this

Which value of $C$ gives the curve through $(1, 4)$?

Example 2●●

Solve $\dfrac{dy}{dx} = 6x^2 - 4x$, given that $y = 5$ when $x = 1$.

Integrate: $y = 2x^3 - 2x^2 + C$.

Substitute $(1, 5)$: $5 = 2 - 2 + C$, so $C = 5$ and $y = 2x^3 - 2x^2 + 5$.

Example 3: unknown constants●●●

Find $b$ and $c$ so that $y = e^{3x} + bx + c$ is a solution of $\dfrac{dy}{dx} = 3y - 18x$.

The left side is $3e^{3x} + b$. The right side is $3e^{3x} + 3bx + 3c - 18x$.

Equate coefficients of $x$: $0 = 3b - 18$, so $b = 6$. Equate constants: $b = 3c$, so $c = 2$.

Flashcards

Order of a DE

The order of the highest derivative in it.

General solution

A family of solutions containing arbitrary constants.

Particular solution

The single solution chosen by the initial conditions.

Practice

What is the order of $\dfrac{d^2y}{dx^2} + 3\dfrac{dy}{dx} = x$?
$y = e^{3x}$ is a solution of $\dfrac{dy}{dx} = ky$. Find $k$.
$y = x^2 + C$ passes through $(1, 4)$. Find $C$.
Solve $\dfrac{dy}{dx} = 6x^2$ with $y = 1$ when $x = 0$, then find $y$ when $x = 1$.
In Example 3, find $b$.
How many arbitrary constants does the general solution of a second-order DE usually have?
Which is a solution of $\dfrac{d^2y}{dx^2} + y = 0$?
The initial condition in a DE problem is used to: