\( \newcommand{\sech}{\mathrm{sech} } \) \( \newcommand{\csch}{\mathrm{csch} } \) \( \newcommand{\arcsinh}{\mathrm{arcsinh} } \) \( \newcommand{\arccosh}{\mathrm{arccosh} } \)
17calculus 17calculus
First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Linear
Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
Constant Coefficients
Substitution
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Wronskian
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Resonance
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

17calculus > differential equations > practice

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Differential Equations Alpha List

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Differential Equations Practice Problems

This page contains practice problems solving differential equations. You can find practice problems on each page to work as you are learning a specific technique. However, on exams, you will not usually be told what technique to use and that is the fun part of differential equations, deciding what technique to use. So, this page is provided with various types of problems but we don't tell you what technique to use. This should help you prepare for your exams.

If you are learning a specific group of techniques, like all second-order, you can use the filter to show only the problems in the group you want to see. Of course, you can also use the filters to show only one type of problem but it might help you to go the page with that specific technique to get more help.

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second-order practice filters

constant coefficients, homogeneous

reduction of order

undetermined coefficients

variation of parameters

first-order practice filters

separable

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exact

Practice Problems

Instructions
1. Unless otherwise instructed, find the general solution to each differential equation.
2. If initial conditions are given, find the particular solution also.
3. If one solution is given, find the second solution and the general solution.
4. Give your answers in exact terms and completely factored.

Level A - Basic

Practice A01

\(\displaystyle{\frac{dy}{dx}=xy}\)

solution

Practice A02

\(\displaystyle{\frac{dy}{dx}=y^2(1+x^2)}\); \(y(0)=1\)

solution

Practice A03

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

answer

solution

Practice A04

\(y''-5y'+6y=4e^t\)

answer

solution

Practice A05

\(y''+3y'-4y=2e^x\); \(y(0)=1, y'(0)=2\)

answer

solution

Practice A06

\(\displaystyle{y\frac{dy}{dx}=x^2+\sech^2(x)}\); \(y(0)=2\)

solution

Practice A07

\(y''-4y'+4y=0\)

answer

solution

Practice A08

\(y''-3y'-4y=3e^{2x}\)

answer

solution

Practice A09

\(\displaystyle{\frac{y+2}{x^2-x+2}\frac{dy}{dx}=\frac{x}{y}}\); \(y(1)=2\)

solution

Practice A10

\(\displaystyle{\frac{dy}{dx}+\frac{2x}{1+x^2}y=\frac{4}{(1+x^2)^2}}\)

answer

solution

Practice A11

\(\displaystyle{\frac{dy}{dx}+y/x=x}\); \(x > 0\); \(y(1)=0\)

answer

solution

Practice A12

\(\displaystyle{\frac{dy}{dx}+3y=2xe^{-3x}}\)

answer

solution

Practice A13

\((y\cos(x)+2xe^y)+(\sin(x)+x^2e^y-1)y'=0\)

answer

solution

Practice A14

\(x^2y''+5xy'-5y=0\), \(x > 0\); \(y_1=x\)

solution

Practice A15

\(x^2y''-3xy'+4y=0\); \(y_1=x^2\)

solution

Practice A16

\(\displaystyle{\frac{dy}{dx}\cdot\frac{y^3+y}{x^2+3x}=1}\)

solution

Practice A17

\(t^2y''-t(t+2)y'+(t+2)y=0\); \(y_1=t\)

solution

Practice A18

\(2xy~dx+(x^2+3y^2)~dy=0\)

solution

Practice A19

\(\ddot{x}+\dot{x}=t^4\)

answer

solution

Practice A20

\(x^2y''+3xy'+y=0\); \(y_1=1/x\)

solution

Practice A21

\(4y''-4y'-3y=0\)

answer

solution

Practice A22

\(y''-18y'+77y=0\); \(y(0)=4, y'(0)=8\)

answer

solution

Practice A23

\(y''+4y'+20y=0\); \(y(0)=9, y'(0)=10\)

answer

solution

Practice A24

\(\dot{x}+3x=t^2+t\)

answer

solution

Practice A25

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

solution

Practice A26

\([\cos(x)\sin(x)-xy^2]~dx+y(1-x^2)~dy=0\)

answer

solution

Practice A27

\(xy'=y+x^2\sin(x)\); \(y(\pi)=0\)

answer

solution

Practice A28

\(\displaystyle{\frac{dy}{dx}=e^{4x-y}}\); \(y(0)=5\)

solution

Practice A29

\(\displaystyle{\frac{dy}{dx}=\cos(x)}\); \(y(0)=-1\)

solution

Practice A30

\(\displaystyle{\frac{dy}{dx}=x/y^2}\)

solution

Practice A31

\(\displaystyle{\frac{dy}{dx}=2x\sqrt{y-1}}\)

solution

Practice A32

\(\displaystyle{\frac{dy}{dx}-2xy=x}\)

answer

solution

Practice A33

\(2xy~dx+x^2~dy=0\)

solution

Practice A34

\(\displaystyle{\frac{dy}{dx}=\frac{x^2+1}{x^2(3y^2+1)}}\)

solution

Practice A35

\(\displaystyle{\frac{dy}{dx}=\frac{y\cos(x)}{1+2y^2}}\); \(y(0)=1\)

solution

Practice A36

\(y''-4y'-5y=0\); \(y(0)=1, y'(0)=0\)

answer

solution

Practice A37

\(y''+22y'+121y=0\); \(y(0)=2, y'(0)=-25\)

answer

solution

Practice A38

\(y''-y'-6y=0\)

answer

solution

Practice A39

\(2x+3+(2y-2)y'=0\)

answer

solution

Practice A40

\(2xy~dx+(x^2-1)~dy=0\)

answer

solution

Practice A41

\(\displaystyle{t^2\frac{dy}{dt}+2ty=\sin(t)}\)

answer

solution

Practice A42

\(y''-2y'+5y=0\)

answer

solution

Practice A43

\(\displaystyle{\frac{du}{dt}=\frac{2t+\sec^2(t)}{2u}}\); \(u(0)=-5\)

solution

Practice A44

\(2y''-11y'+12y=0\); \(y(0)=5, y'(0)=15\)

answer

solution

Practice A45

\(y''+y'-6y=0\)

answer

solution

Practice A46

\(y''-8y'+16y=0\)

answer

solution

Practice A47

\(y''-6y'+13y=0\)

answer

solution

Practice A48

\(y''-7y'+10y=0\)

answer

solution

Practice A49

\(y''-5y'+6y=4e^t\)

answer

solution

Practice A50

\(\displaystyle{\frac{xy'-y}{x^2}=0}\)

answer

solution

Practice A51

\(y''+6y'+9y=0\)

answer

solution

Practice A52

\(y''-5y'+6y=12e^{5x}\)

answer

solution

Practice A53

\(\displaystyle{\frac{dy}{dx}-2y=e^{3x}}\)

answer

solution

Practice A54

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

solution

Practice A55

\(\displaystyle{\frac{dy}{dx}=\frac{4-2x}{3y^2-5}}\); \(y(1)=3\)

solution

Practice A56

\(y''+3y'+2y=x^2\)

answer

solution

Practice A57

\(\displaystyle{\frac{dy}{dx}=2\sqrt{y}}\); \(y(0)=9\)

solution

Practice A58

\(y''+20y'+100y=0\)

answer

solution

Practice A59

\(y''-5y'+6y=2x+3\)

answer

solution


Level B - Intermediate

Practice B01

\(\displaystyle{y''-2y'+y=\frac{e^x}{x^4}}\)

answer

solution

Practice B02

\(y''+2y'=24x+e^{-2x}\)

answer

solution

Practice B03

\(y''+4y'+4y=130\cos(3x)\)

answer

solution

Practice B04

\(dv/ds=(s+1)/(sv+s)\)

answer

solution

Practice B05

\((3x+y+1)~dx+(3y+x+1)~dy=0\)

solution

Practice B06

\(\displaystyle{y''-2y'+y=\frac{e^x}{x^2+1}}\)

answer

solution

Practice B07

\(y''+4y=\csc(2x)\)

answer

solution

Practice B08

\(\displaystyle{\frac{dy}{dx}=\frac{3x^2+4x+2}{2(y-1)}}\); \(y(0)=-1\)

solution

Practice B09

\((2x+y+1)~dx+(2y+x+1)~dy=0\)

solution

Practice B10

\(y''-5y'+6y=10e^{2x}\)

answer

solution

Practice B11

\(\displaystyle{y''+y=\frac{1}{\cos(x)}}\)

answer

solution

Practice B12

\(y''-3y'-4y=2\sin(x)\)

answer

solution

Practice B13

\(y''-3y'-4y=4x^2\)

answer

solution

Practice B14

\(y''+y=\tan(x)\)

answer

solution

Practice B15

\((3x^2-2xy+2)~dx+(6y^2-x^2+3)~dy=0\)

solution


Level C - Advanced

Practice C01

\(y''+9y=2\tan(3x)\)

answer

solution

Practice C02

\(y''+3y'+2y=5\cos(t)\)

answer

solution

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