Introduction to Differential Equations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Introduction to Differential Equations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An equation that contains an unknown function and one or more of its derivatives. Solving a DE means finding the function(s) that satisfy the equation.

An algebraic equation like $x^2 = 4$ asks 'what number satisfies this?' A differential equation like $\frac{dy}{dx} = 2x$ asks 'what function has this derivative?' The answer isn't a number but a family of functions: $y = x^2 + C$ .

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A differential equation relates a function to its derivatives; solving it means finding the function(s) that fit.

Common stuck point: The procedure for introduction to differential equations is the easy part; the trap is dropping the constant of integration. Asking "Does the equation involve an unknown function together with its derivative(s), with the goal of finding that function?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the equation involve an unknown function together with its derivative(s), with the goal of finding that function?

Worked Examples

Example 1

easy

Verify

y = 3e^{2x}

solves

y' = 2y

, and find the particular solution with

y(0) = 5

Answer

Verified; particular solution

y = 5e^{2x}

First step

y' = 6e^{2x} = 2(3e^{2x}) = 2y

. ✓

Full solution

2
General: $y = Ce^{2x}$ . Apply $y(0)=5$ : $C=5$ .
3
Particular: $y = 5e^{2x}$ .

Verify by substitution; the initial condition pins down

C

Example 2

medium

Find the general and particular (

y(0)=4

) solution to

y' = 3x^2+1

Example 3

easy

Verify that

y = e^{-x}

satisfies

y' + y = 0

Example 4

medium

Solve the IVP

y'' = 2

y(0)=1

y'(0)=3

Example 5

hard

Verify

y = 2\cos(3x) - \sin(3x)

solves

y'' + 9y = 0

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Verify that

y = \sin x + 2\cos x

satisfies

y'' + y = 0

Example 2

medium

Find the particular solution to

y' = -2y

with

y(0)=3

Example 3

easy

What is the order of

y''+3y'-y=0

Example 4

easy

Solve

\frac{dy}{dx}=2x

for the general solution.

Example 5

easy

y=e^{2x}

a solution of

y'=2y

Example 6

easy

Why does

\frac{dy}{dx}=3x^2

have infinitely many solutions?

Example 7

easy

What is the degree of

(y')^3+y=x

Example 8

easy

Find the particular solution of

\frac{dy}{dx}=4

with

y(0)=3

Example 9

easy

Classify

\frac{dy}{dx}=ky

in words.

Example 10

easy

\frac{dy}{dx}=x+y

separable as stated?

Example 11

medium

Solve the IVP

\frac{dy}{dx}=6x^2

y(1)=5

Example 12

medium

Verify

y=C_1\cos x+C_2\sin x

solves

y''+y=0

Example 13

medium

Solve

\frac{dy}{dx}=\cos x

with

y(\pi/2)=0

Example 14

medium

How many arbitrary constants does the general solution of a 3rd-order DE have?

Example 15

medium

y=x^2

a solution of

xy'=2y

Example 16

medium

Find the general solution of

y''=6x

Example 17

medium

For exponential decay

\frac{dy}{dt}=-0.5y

, write the general solution.

Example 18

medium

Determine the order and degree of

\left(\frac{d^2y}{dx^2}\right)^2+y'=x

Example 19

medium

Solve the IVP

\frac{dy}{dx}=3x^2-2

y(1)=4

Example 20

challenge

Find the particular solution of

y''=12x^2

with

y(0)=1

y'(0)=2

Example 21

challenge

Show that

y=\frac{1}{x}

solves

y'=-y^2

and state where it is valid.

Example 22

challenge

Find the value of

r

so that

y=e^{rx}

solves

y''-5y'+6y=0

Example 23

easy

State the order of

y''' - y' = e^x

Example 24

easy

Find the general solution of

\dfrac{dy}{dx} = 5

Example 25

easy

y = e^{-3x}

a solution of

y' + 3y = 0

Example 26

easy

Solve

\dfrac{dy}{dx} = e^x

for the general solution.

Example 27

medium

Solve the IVP

\dfrac{dy}{dx} = e^x

y(0) = 2

Example 28

medium

Find the particular solution of

\dfrac{dy}{dx} = \dfrac{1}{x}

with

y(1) = 0

Example 29

medium

Find

k

so that

y = e^{kt}

satisfies

\dfrac{dy}{dt} = 7y

Example 30

medium

A bacteria population satisfies

\dfrac{dP}{dt} = 0.2 P

with

P(0) = 50

. Find

P(t)

Example 31

medium

Solve

y'' = 0

for its general solution.

Example 32

medium

Determine the order of

\left(\dfrac{d^3 y}{dx^3}\right) + \sin y = 0

Example 33

medium

y = x e^x

a solution of

y' - y = e^x

Example 34

medium

Write a DE whose general solution is

y = Ce^{4x}

Example 35

hard

Find the particular solution of

\dfrac{dy}{dt} = -3y

with

y(0) = 8

, then find

y(2)

Example 36

hard

Find all

r

so that

y = e^{rx}

solves

y'' + y' - 6y = 0

Example 37

hard

Solve

y' = y

y(0) = -2

Example 38

hard

Solve the IVP

y' = 2xy

y(0) = 3

by inspection (using

y = Ce^{x^2}

Example 39

hard

Newton's cooling:

\dfrac{dT}{dt} = -k(T - 20)

with

T(0) = 90

. Write the general form of

T(t)

Example 40

hard

State the order and degree of

\left(\dfrac{dy}{dx}\right)^4 + y = 0

Example 41

hard

Show that

y = x^2 + \dfrac{C}{x}

solves

xy' + y = 3x^2

for any constant

C

Example 42

challenge

Solve

y'' - 4y = 0

for the general real solution.

Example 43

challenge

Find the value of

r

so that

y = x^r

solves

x^2 y'' - 2y = 0

← Back to Introduction to Differential Equations Formula Explained Practice Mode

Related Concepts

Derivative Integral Slope Fields Separation of Variables

Background Knowledge

These ideas may be useful before you work through the harder examples.

derivativeintegral