Exercises: Linear and Separable ODEs

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\bigmath }[1]{$\displaystyle #1$} \newcommand {\choicebreak }[0]{} \newcommand {\type }[0]{} \newcommand {\notes }[0]{} \newcommand {\keywords }[0]{} \newcommand {\offline }[0]{} \newcommand {\comments }[0]{\begin {feedback}} \newcommand {\multiplechoice }[0]{\begin {multipleChoice}} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

Exercises related to solving linear and separable ODEs.

Remember: several of these exercises involve logarithms. Use absolute value signs inside the logarithm when they’re needed.

Find the general solution of the ODE below. $y' = y - 2$ $\text {General Solution: } \ \ y = C \answer {e^x} + \answer {2}$ (For definiteness, the expression you enter in the second blank should equal $2$ when $x = 0$ .)

This ODE is both linear and separable, so either approach will work.

Find the general solution of the ODE below. $x^2 y' + xy = 1$ $\text {General Solution: } \ \ y = \answer {\frac {\ln |x|}{x}} + \frac {C}{x}$ (For definiteness, the function you enter for your answer should equal $0$ at $x=1$ . If that’s not the case, you might need to rewrite your solution and redefine the constant $C$ .)

This is a linear ODE.

Find the general solution of the ODE below. $y y' = 4x$ $\text {General Solution: } \ \ y^2 = \answer {4 x^2} + C$ (For definiteness, the function you enter for your answer should equal $0$ at $x=0$ . If that’s not the case, you might need to rewrite your solution and redefine the constant $C$ .)

This is a separable ODE. In the form written, it’s already separated.

$y'-3y=xe^{2x}$ $\text {General Solution: } \ \ y = C\answer {e^{3x}}-(x+1)\answer {e^{2x}}$

This is a linear ODE.

$(x^2+1) y' = \frac {x}{y-1}$ $\text {General Solution: } \ \ (y-1)^2 = \answer {\ln (x^2 + 1)} + C$ (For definiteness, the function you enter for your answer should equal $0$ at $x=0$ . If that’s not the case, you might need to rewrite your solution and redefine the constant $C$ .)

This is a separable ODE.

Solve the initial value problem $y' = \cos ^2 x \cos ^2 2y \text { with } y(0) = 0.$ $\text {Solution: } \ \ y = \answer {\frac {1}{2} \arctan \left ( x + \frac {1}{2} \sin 2x \right )}$

This is a separable ODE.

Solve the initial value problem $y' + (\tan x) y = \sec x \text { with } y(0) = -3.$ $\text {Solution: } \ \ y = \answer {\sin x - 3\cos x}$

This is a linear ODE. The integral of $\tan x$ is $-\ln |\cos x|$ .

Sample Quiz Questions

Let $y(x)$ be the solution to the initial value problem $\frac {dy}{dx} = -(1 + 3 x^2)y^2$ and $y(0) = 1/2$ . What is the value of $y(1)$ ?

This is a separable ODE. Moving all functions of $y$ to the left-hand side and all functions of $x$ to the right-hand side and integrating gives $\int \frac { -1 }{ y^2} ~ dy = \int (1 + 3 x^2) ~ dx,$ which yields $\frac {1}{y} = x^3 + x + C.$ Evaluating at $x = 0$ and $y = 1/2$ gives $2 = 0 + C$ , so $\frac {1}{y} = x^3 + x + 2,$ i.e., $y = \frac {1}{x^3 + x + 2}.$ Plugging in $x = 1$ gives $y = 1/4$ .

Sample Exam Questions

The solution of the initial value problem $\displaystyle x \frac {dy}{dx} + 3y = 7 x^4$ , $y(1) = 1$ , satisfies $y(2) =$

The solution of the initial value problem $\displaystyle \frac {dy}{dx} - 20 x^4 e^{-y} = 0$ , $y(0) = 0$ , satisfies $y(1) =$

Let $y(x)$ be the solution of the initial value problem $x \frac {dy}{dx} = e^x - y \ \text { with } \ y(\ln 2) = 0.$ Find $y(1)$ .

Let $y(x)$ be the solution of the initial value problem $x \frac {dy}{dx} = y + x^2 \sin x \ \text { with } \ y (\pi ) = 0.$ What is $y(2 \pi )$ ?

Consider the initial value problem $(1 + x^2) \frac {dy}{dx} = 2y \ \text { with } \ y(0) = 2.$ What is $\displaystyle \lim _{x \rightarrow \infty } y(x)$ ?

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Sample Quiz Questions

Sample Exam Questions

Controls

Symbols

Settings