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 x2=4x^2 = 4 asks 'what number satisfies this?' A differential equation like dydx=2x\frac{dy}{dx} = 2x asks 'what function has this derivative?' The answer isn't a number but a family of functions: y=x2+Cy = x^2 + C.

Showing a random 20 of 50 problems.

Example 1

easy
Is y=e−3xy = e^{-3x} a solution of y′+3y=0y' + 3y = 0?

Example 2

easy
What is the degree of (y′)3+y=x(y')^3+y=x?

Example 3

challenge
Show that y=1xy=\frac{1}{x} solves y′=−y2y'=-y^2 and state where it is valid.

Example 4

medium
For exponential decay dydt=−0.5y\frac{dy}{dt}=-0.5y, write the general solution.

Example 5

medium
Find the particular solution of dydx=1x\dfrac{dy}{dx} = \dfrac{1}{x} with y(1)=0y(1) = 0.

Example 6

hard
Solve the IVP y′=2xyy' = 2xy, y(0)=3y(0) = 3 by inspection (using y=Cex2y = Ce^{x^2}).

Example 7

hard
Show that y=x2+Cxy = x^2 + \dfrac{C}{x} solves xy′+y=3x2xy' + y = 3x^2 for any constant CC.

Example 8

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

Example 9

medium
Solve the IVP dydx=ex\dfrac{dy}{dx} = e^x, y(0)=2y(0) = 2.

Example 10

easy
Is y=e2xy=e^{2x} a solution of y′=2yy'=2y?

Example 11

medium
A bacteria population satisfies dPdt=0.2P\dfrac{dP}{dt} = 0.2 P with P(0)=50P(0) = 50. Find P(t)P(t).

Example 12

medium
Find the general solution of y′′=6xy''=6x.

Example 13

medium
Solve the IVP y′′=2y'' = 2, y(0)=1y(0)=1, y′(0)=3y'(0)=3.

Example 14

medium
Solve the IVP dydx=6x2\frac{dy}{dx}=6x^2, y(1)=5y(1)=5.

Example 15

hard
Solve y′=yy' = y, y(0)=−2y(0) = -2.

Example 16

easy
Write the DE that says 'the rate of change of yy with respect to tt is proportional to yy.'

Example 17

medium
Determine the order of (d3ydx3)+sin⁡y=0\left(\dfrac{d^3 y}{dx^3}\right) + \sin y = 0.

Example 18

easy
Verify y=3e2xy = 3e^{2x} solves y′=2yy' = 2y, and find the particular solution with y(0)=5y(0) = 5.

Example 19

hard
Find all rr so that y=erxy = e^{rx} solves y′′+y′−6y=0y'' + y' - 6y = 0.

Example 20

easy
Classify dydx=ky\frac{dy}{dx}=ky in words.
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