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

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Differential Equations Alpha List

 Boundary Value Problems Cauchy-Euler Equations Chebyshev Equations Chemical Concentration Classify Differential Equations Constant Coefficients Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay First Order, Linear Fluids (Mixing) Fourier Series Inhomogeneous ODE's Integrating Factors (Exact) Integrating Factors (Linear) Laplace Transforms Linear Systems Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance Second Order, Linear Separation of Variables Shifting Theorems Slope Fields Solve Initial Value Problems Square Wave Substitution Undetermined Coefficients Unit Impulse Function Unit Step Function Variation of Parameters Wronskian

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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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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.

 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

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

solution

Practice A04

$$y''-5y'+6y=4e^t$$

solution

Practice A05

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

solution

Practice A06

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

solution

Practice A07

$$y''-4y'+4y=0$$

solution

Practice A08

$$y''-3y'-4y=3e^{2x}$$

solution

Practice A09

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

solution

Practice A10

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

solution

Practice A11

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

solution

Practice A12

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

solution

Practice A13

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

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

$$\ddot{x}+\dot{x}=t^4$$

solution

Practice A20

$$x^2y''+3xy'+y=0$$; $$y_1=1/x$$

solution

Practice A21

$$4y''-4y'-3y=0$$

solution

Practice A22

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

solution

Practice A23

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

solution

Practice A24

$$\dot{x}+3x=t^2+t$$

solution

Practice A25

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

solution

Practice A26

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

solution

Practice A27

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

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

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

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

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

solution

Practice A37

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

solution

Practice A38

$$y''-y'-6y=0$$

solution

Practice A39

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

solution

Practice A40

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

solution

Practice A41

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

solution

Practice A42

$$y''-2y'+5y=0$$

solution

Practice A43

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

solution

Practice A44

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

solution

Practice A45

$$y''+y'-6y=0$$

solution

Practice A46

$$y''-8y'+16y=0$$

solution

Practice A47

$$y''-6y'+13y=0$$

solution

Practice A48

$$y''-7y'+10y=0$$

solution

Practice A49

$$y''-5y'+6y=4e^t$$

solution

Practice A50

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

solution

Practice A51

$$y''+6y'+9y=0$$

solution

Practice A52

$$y''-5y'+6y=12e^{5x}$$

solution

Practice A53

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

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

$$y''+3y'+2y=x^2$$

solution

Practice A57

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

solution

Practice A58

$$y''+20y'+100y=0$$

solution

Practice A59

$$y''-5y'+6y=2x+3$$

solution

 Level B - Intermediate

Practice B01

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

solution

Practice B02

$$y''+2y'=24x+e^{-2x}$$

solution

Practice B03

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

solution

Practice B04

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

solution

Practice B05

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

solution

Practice B06

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

solution

Practice B07

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

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

$$y''-5y'+6y=10e^{2x}$$

solution

Practice B11

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

solution

Practice B12

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

solution

Practice B13

$$y''-3y'-4y=4x^2$$

solution

Practice B14

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

solution

Practice B15

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

solution

Practice C01

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

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