# [SOLVED]2.1.9 Find the general solution of the given differential equation

#### karush

##### Well-known member
$\tiny{2.1.9}$
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     ------------------------------------
$\color{red}\displaystyle y=ce^{-t/2}+3t-6 \\ \textit{y is asymptotic to } 3t-6 \textit{ as } t\to\infty$

#### MarkFL

Staff member
When you determine your integrating factor, you need only determine up to but not including the constant of integration:

$$\displaystyle \mu(t)=\exp\left(\int\frac{1}{2}\,dt\right)=e^{\frac{t}{2}}$$

And so the ODE becomes:

$$\displaystyle e^{\frac{t}{2}}y'+\frac{1}{2}e^{\frac{t}{2}}y=\frac{3}{2}te^{\frac{t}{2}}$$

Or:

$$\displaystyle \frac{d}{dt}\left(e^{\frac{t}{2}}y\right)=\frac{3}{2}te^{\frac{t}{2}}$$

Integrating, we obtain:

$$\displaystyle e^{\frac{t}{2}}y=3e^{\frac{t}{2}}(t-2)+c_1$$

Now does the answer given by the book make sense? Hence:

$$\displaystyle y(t)=3(t-2)+c_1e^{-\frac{t}{2}}$$

#### karush

##### Well-known member
for some reason I thought $\displaystyle e^{t/2}$ stayed the same whether taking the integral or derivative

$\displaystyle\frac{d}{dt} e^{t/2} = \frac{e^{t/2}}{2}$

multiply thru
$\displaystyle e^{t/2}y'+\frac{1}{2}e^{t/2}y =\left(e^{t/2}y\right)^\prime =\frac{3}{2}te^{t/2}$
integrate through
$\displaystyle e^{t/2}y =\int\frac{3}{2}te^{t/2}\ dt=3 e^{t/2}(t-2)+c$
simplify
$\displaystyle y=ce^{t/2}+3t-6$
=========================================
$\textit{ok I didn't know how this the book says y is asymptotic to$3t-6$as$t\to\infty$}$

Draw a direction field for the given differential equation.
how do you do this with Desmos

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Staff member

#### topsquark

##### Well-known member
MHB Math Helper
$\displaystyle e^{t/2}y =3 e^{t/2}(t-2)+c$
simplify
$\displaystyle y=ce^{t/2}+3t-6$
Think about these two lines for a moment...

-Dan

#### karush

##### Well-known member
Think about these two lines for a moment...

-Dan
$\displaystyle y=3(t-2)+\frac{c}{e^{t/2}}=ce^{-t/2}+3t-6$

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#### topsquark

##### Well-known member
MHB Math Helper
$\displaystyle y=3(t-2)+\frac{c}{e^{t/2}}=ce^{-t/2}+3t-6$
You just aren't doing well with the typos today, huh? Whatever. You got it.

-Dan

#### karush

##### Well-known member
I just wanted to see if you would catch it... 