Practice Separation of Variables in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

A method for solving first-order DEs of the form $\frac{dy}{dx} = f(x) \cdot g(y)$ : rearrange to $\frac{dy}{g(y)} = f(x)\,dx$ , then integrate both sides.

If the rate of change factors into a piece that depends only on $x$ and a piece that depends only on $y$ , you can sort them onto opposite sides of the equation—all the $y$ -stuff on the left, all the $x$ -stuff on the right—then integrate each side in its own variable.

Showing a random 20 of 50 problems.

Example 1

medium

Solve the IVP

\frac{dy}{dx} = \frac{x+1}{y}

y(0)=2

Example 2

easy

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

separable?

Example 3

medium

Solve

\frac{dy}{dx}=e^{x-y}

Example 4

challenge

A skydiver's velocity satisfies

\frac{dv}{dt}=g - kv

with

v(0)=0

g=9.8

m/s

^2

k=0.2

^{-1}

. Solve for

v(t)

and find the terminal velocity.

Example 5

medium

Solve

\frac{dy}{dx}=\frac{y^2-1}{x}

for

x>0

y(1)=0

, by separation. Identify the constant.

Example 6

medium

Solve

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

with

y(0)=1

Example 7

challenge

Solve

\frac{dy}{dx}=\frac{2x}{1+y^2}

with

y(0)=0

Example 8

medium

Find equilibrium solutions of

\frac{dy}{dx} = (y-3)(y+1)

Example 9

easy

Separate the variables in

\frac{dy}{dx}=xy

Example 10

easy

\frac{dy}{dx}=x^2+y^2

separable?

Example 11

medium

Solve

\frac{dy}{dx}=y^2

with

y(0)=1

Example 12

medium

Solve

\frac{dy}{dx}=ky

generally by separation.

Example 13

medium

A radioactive substance decays with

\frac{dN}{dt}=-\lambda N

. Solve for

N(t)

given

N(0)=N_0

Example 14

medium

Solve

\frac{dy}{dx}=e^{x}\cdot e^{-y}

with

y(0)=0

Example 15

easy

Solve

dy/dx = xy

with

y(0)=2

Example 16

easy

After separating

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

, integrate both sides.

Example 17

challenge

Solve

\frac{dy}{dx} = \frac{2xy}{x^2+1}

generally.

Example 18

medium

Find the general solution to

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

Example 19

medium

Solve

\frac{dy}{dx}=y^2 \sec^2 x

with

y(0)=1

Example 20

challenge

Solve

\frac{dy}{dx}=\frac{y(1-y)}{x}

for

x>0

, with

y(1)=1/2

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Related Concepts

Introduction to Differential Equations Integral