Practice Introduction to Differential Equations in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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: $y = x^2 + C$ .

Showing a random 20 of 50 problems.

Example 1

easy

y = e^{-3x}

a solution of

y' + 3y = 0

Example 2

easy

What is the degree of

(y')^3+y=x

Example 3

challenge

Show that

y=\frac{1}{x}

solves

y'=-y^2

and state where it is valid.

Example 4

medium

For exponential decay

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

, write the general solution.

Example 5

medium

Find the particular solution of

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

with

y(1) = 0

Example 6

hard

Solve the IVP

y' = 2xy

y(0) = 3

by inspection (using

y = Ce^{x^2}

Example 7

hard

Show that

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

solves

xy' + y = 3x^2

for any constant

C

Example 8

medium

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

Example 9

medium

Solve the IVP

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

y(0) = 2

Example 10

easy

y=e^{2x}

a solution of

y'=2y

Example 11

medium

A bacteria population satisfies

\dfrac{dP}{dt} = 0.2 P

with

P(0) = 50

. Find

P(t)

Example 12

medium

Find the general solution of

y''=6x

Example 13

medium

Solve the IVP

y'' = 2

y(0)=1

y'(0)=3

Example 14

medium

Solve the IVP

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

y(1)=5

Example 15

hard

Solve

y' = y

y(0) = -2

Example 16

easy

Write the DE that says 'the rate of change of

y

with respect to

t

is proportional to

y

Example 17

medium

Determine the order of

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

Example 18

easy

Verify

y = 3e^{2x}

solves

y' = 2y

, and find the particular solution with

y(0) = 5

Example 19

hard

Find all

r

so that

y = e^{rx}

solves

y'' + y' - 6y = 0

Example 20

easy

Classify

\frac{dy}{dx}=ky

in words.

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Related Concepts

Derivative Integral Slope Fields Separation of Variables