Separation of Variables Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Separation of Variables.

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

Concept 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.

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: If $\frac{dy}{dx}$ factors into an $x$ -part times a $y$ -part, separate them onto opposite sides and integrate each.

Common stuck point: The procedure for separation of variables is the easy part; the trap is trying to separate a non-product right side. Asking "Can I rewrite the DE so one side has only $y$ and $dy$ , the other only $x$ and $dx$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I rewrite the DE so one side has only $y$ and $dy$ , the other only $x$ and $dx$ ?

Worked Examples

Example 1

easy

Solve

dy/dx = xy

with

y(0)=2

Answer

y = 2e^{x^2/2}

First step

Separate:

dy/y = x\,dx

. Integrate:

\ln|y| = x^2/2+C

Full solution

2
$y = Ae^{x^2/2}$ . $y(0)=2 \Rightarrow A=2$ .
3
Solution: $y = 2e^{x^2/2}$ .

Separate, integrate both sides, exponentiate, apply IC.

Example 2

hard

Solve the logistic DE

dy/dt = y(1-y)

with

y(0) = 1/2

Example 3

easy

Solve

\frac{dy}{dx} = 2x

with

y(0)=5

Example 4

medium

A radioactive substance decays with

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

. Solve for

N(t)

given

N(0)=N_0

Example 5

hard

A tank initially holds

50

L of pure water. Brine with

0.2

kg/L of salt flows in at

2

L/min, and the well-mixed solution flows out at the same rate. Let

S(t)

be salt amount; derive and solve the ODE.

Example 6

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.

Practice Problems

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

Example 1

easy

Solve

dy/dx = x/y

with

y(0)=3

Example 2

medium

Solve Newton's cooling:

dT/dt = -0.1(T-20)

T(0)=80

Example 3

easy

\frac{dy}{dx}=xy

separable?

Example 4

easy

Separate the variables in

\frac{dy}{dx}=xy

Example 5

easy

Solve

\frac{dy}{dx}=y

by separation.

Example 6

easy

After separating

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

, integrate both sides.

Example 7

easy

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

separable?

Example 8

easy

What equilibrium solution might be lost when solving

\frac{dy}{dx}=y

by dividing by

y

Example 9

easy

Separate

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

Example 10

easy

Solve

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

Example 11

medium

Solve the IVP

\frac{dy}{dx}=xy

y(0)=2

Example 12

medium

Solve

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

with

y(0)=3

Example 13

medium

Solve

\frac{dy}{dx}=ky

generally by separation.

Example 14

medium

Solve

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

with

y(0)=1

Example 15

medium

Solve Newton's cooling

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

generally.

Example 16

medium

Solve

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

with

y(0)=1

Example 17

medium

Find the equilibrium solutions of

\frac{dy}{dx}=y(2-y)

before separating.

Example 18

medium

Solve

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

Example 19

medium

Solve

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

for

x>0

by separation.

Example 20

challenge

Solve the logistic IVP

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

y(0)=\frac12

Example 21

challenge

Solve

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

with

y(0)=0

Example 22

challenge

A tank's salt obeys

\frac{dS}{dt}=-\frac{S}{100}

with

S(0)=20

. Find

S(t)

and the half-life.

Example 23

easy

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

separable?

Example 24

easy

Solve

\frac{dy}{dx}=3

by separation, with

y(0)=2

Example 25

easy

Solve

\frac{dy}{dx}=-3y

with

y(0)=10

Example 26

medium

Solve

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

with

y(0)=1

Example 27

medium

Solve

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

for

x>0

y(1)=0

, by separation. Identify the constant.

Example 28

medium

Solve

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

with

y(0)=0

Example 29

medium

Solve the IVP

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

y(0)=2

Example 30

medium

Solve

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

with

y(0)=1

Example 31

medium

Find the general solution to

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

Example 32

medium

Solve Newton's heating:

\frac{dT}{dt} = k(100-T)

with

T(0)=20

k=0.05

Example 33

medium

Solve

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

with

y(1)=e

Example 34

hard

Solve

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

with

y(0)=1

Example 35

hard

Solve

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

with

y(0)=2

Example 36

hard

Solve the IVP

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

with

y(0)=0

Example 37

hard

Solve

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

generally.

Example 38

challenge

Solve

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

for

x>0

, with

y(1)=1/2

Example 39

challenge

Solve

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

generally.

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Related Concepts

Introduction to Differential Equations Integral

Background Knowledge

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

differential equations introintegral