Practice Separation of Variables in Math

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

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Quick Recap

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

If the rate of change factors into a piece that depends only on xx and a piece that depends only on yy, you can sort them onto opposite sides of the equationβ€”all the yy-stuff on the left, all the xx-stuff on the rightβ€”then integrate each side in its own variable.

Showing a random 20 of 50 problems.

Example 1

medium
Solve the IVP dydx=x+1y\frac{dy}{dx} = \frac{x+1}{y}, y(0)=2y(0)=2.

Example 2

easy
Is dydx=x+y\frac{dy}{dx}=x+y separable?

Example 3

medium
Solve dydx=exβˆ’y\frac{dy}{dx}=e^{x-y}.

Example 4

challenge
A skydiver's velocity satisfies dvdt=gβˆ’kv\frac{dv}{dt}=g - kv with v(0)=0v(0)=0, g=9.8g=9.8 m/s2^2, k=0.2k=0.2 sβˆ’1^{-1}. Solve for v(t)v(t) and find the terminal velocity.

Example 5

medium
Solve dydx=y2βˆ’1x\frac{dy}{dx}=\frac{y^2-1}{x} for x>0x>0, y(1)=0y(1)=0, by separation. Identify the constant.

Example 6

medium
Solve dydx=cos⁑xβ‹…y\frac{dy}{dx}=\cos x\cdot y with y(0)=1y(0)=1.

Example 7

challenge
Solve dydx=2x1+y2\frac{dy}{dx}=\frac{2x}{1+y^2} with y(0)=0y(0)=0.

Example 8

medium
Find equilibrium solutions of dydx=(yβˆ’3)(y+1)\frac{dy}{dx} = (y-3)(y+1).

Example 9

easy
Separate the variables in dydx=xy\frac{dy}{dx}=xy.

Example 10

easy
Is dydx=x2+y2\frac{dy}{dx}=x^2+y^2 separable?

Example 11

medium
Solve dydx=y2\frac{dy}{dx}=y^2 with y(0)=1y(0)=1.

Example 12

medium
Solve dydx=ky\frac{dy}{dx}=ky generally by separation.

Example 13

medium
A radioactive substance decays with dNdt=βˆ’Ξ»N\frac{dN}{dt}=-\lambda N. Solve for N(t)N(t) given N(0)=N0N(0)=N_0.

Example 14

medium
Solve dydx=exβ‹…eβˆ’y\frac{dy}{dx}=e^{x}\cdot e^{-y} with y(0)=0y(0)=0.

Example 15

easy
Solve dy/dx=xydy/dx = xy with y(0)=2y(0)=2.

Example 16

easy
After separating dyy=x dx\frac{dy}{y}=x\,dx, integrate both sides.

Example 17

challenge
Solve dydx=2xyx2+1\frac{dy}{dx} = \frac{2xy}{x^2+1} generally.

Example 18

medium
Find the general solution to dydx=3x2y2\frac{dy}{dx} = \frac{3x^2}{y^2}.

Example 19

medium
Solve dydx=y2sec⁑2x\frac{dy}{dx}=y^2 \sec^2 x with y(0)=1y(0)=1.

Example 20

challenge
Solve dydx=y(1βˆ’y)x\frac{dy}{dx}=\frac{y(1-y)}{x} for x>0x>0, with y(1)=1/2y(1)=1/2.
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