- #1

karush

Gold Member

MHB

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$\tiny{2.1.9}$

2000

Find the general solution of the given differential equation, and use it to determine how

solutions behave as $t\to\infty$.

$2y'+y=3t$

divide by 2

$y'+\frac{1}{2}y=\frac{3}{2}t$

find integrating factor,

$\displaystyle\exp\left(\int \frac{1}{2} dt\right)=e^{t/2}+c$

multiply thru

$e^{t/2}y'+e^{t/2}\frac{y}{2}

=\frac{3e^{t/2}}{2}t $ok something went

------------------------------------

book answer

$\color{red}\displaystyle y=ce^{-t/2}+3t-6 \\

\textit{y is asymptotic to } 3t-6 \textit{ as } t\to\infty $

2000

Find the general solution of the given differential equation, and use it to determine how

solutions behave as $t\to\infty$.

$2y'+y=3t$

divide by 2

$y'+\frac{1}{2}y=\frac{3}{2}t$

find integrating factor,

$\displaystyle\exp\left(\int \frac{1}{2} dt\right)=e^{t/2}+c$

multiply thru

$e^{t/2}y'+e^{t/2}\frac{y}{2}

=\frac{3e^{t/2}}{2}t $ok something went

------------------------------------

book answer

$\color{red}\displaystyle y=ce^{-t/2}+3t-6 \\

\textit{y is asymptotic to } 3t-6 \textit{ as } t\to\infty $

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