Differential Equations: Problems with Solutions

By Prof. Hernando Guzman Jaimes (University of Zulia - Maracaibo, Venezuela)
Problem 1
What is the solution to this differential equation?
$dx+e^{3x}dy=0$
Problem 2
After applying variable separation method to $dy-(y-1)^{2}dx=0$
we get
Problem 3
Solve the differential equation using variable separation
$\dfrac{dy}{dx}+2xy^{2}=0$
Problem 4
Solve: $xy'+y=e^{x}$, where $y(1)=2$
Problem 5
Solve the differential equation
$\dfrac{dy}{dx}=e^{3x+2y}\qquad y(0)=1$
Problem 6
$e^{x}y\dfrac{dy}{dx}=e^{-y}+e^{-2x-y}$

After applying variable separation we get:
Problem 7
Solve the differential equation:
$y\ln x\dfrac{dx}{dy}=(\dfrac{y+1}{x})^{2}$

$y(1)=1$
Problem 8
$\sqrt{1-y^{2}}dx-\sqrt{1-x^{2}}dy=0,$
$y(0)=\dfrac{\sqrt{3}}{2}$
Problem 9
The result of applying variable separation method in
$\csc y$ $dx+\sec ^{2}x$ $dy=0$ is:
Problem 10
$\sin 3x$ $dx+2y$ $\cos ^{3}3x$ $dy=0$
$y(\pi )=0$

Problem 11
Find the general solution to the differential equation
$(e^{y}+1)^{2}e^{-y}dx+(e^{x}+1)^{3}e^{-x}dy=0$
Problem 12
$x(1+y^{2})^{1/2}dx=y(1+x^{2})^{1/2}dy$
Problem 13
What is the solution to the differential equation?
$\dfrac{dy}{dx}+y=e^{3x}$
Problem 14
After applying the linear equation method to $y'+3x^{2}y=x^{2}$ we get
Problem 15
Solve the differential equation
$y'+2xy=x^{3}$
Problem 16
This differential equation is a linear function
$x^{2}y'+xy=1$
What is the solution?
Problem 17
What is the solution to this linear equation?
$y'=2y+x^{2}+5$
Problem 18
Solve this linear differential equation
$x\dfrac{dy}{dx}-y=x^{2}\sin (x)$
Problem 19
Solve this linear differential equation.
$(1+x)\dfrac{dy}{dx}-xy=x+x^{2}$
Problem 20
Solve this linear differential equation
$x^{2}y'+x(x+2)y=e^{x}$
Problem 21
What is the solution to the differential equation $(x+1)\dfrac{dy}{dx}+y=\ln x$ with initial value $y(1)=10$?
Problem 22
What is the solution with initial value $y(0)=-1$?
$y'+(\tan x)y=\cos ^{2}x$
Problem 23
Find the particular solution to the differential equation $\dfrac{dy}{dx}+2xy=f(x),y(0)=2$ where
$f(x)=\left\{ \begin{array}{c} x,\text{ \ }0\leq x<1 \\ 0,\text{ \ \ \ \ \ \ \ \ }x\geq 1 \end{array} \right\}$
Problem 24
Find the particular solution to the differential equation $(1+x^{2})\frac{dy}{dx}+2xy=f(x),y(0)=0,$ where
$f(x)=\left\{ \begin{array}{c} x,\text{ \ }0\leq x<1 \\ 0,\text{ \ \ \ \ \ \ \ \ }x\geq 1 \end{array} \right\} $
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