Toronto Math Forum
MAT2442018F => MAT244Tests => Quiz1 => Topic started by: Victor Ivrii on September 28, 2018, 03:36:35 PM

Find the solution of the given initial value problem
\begin{equation*}
t^3y' + 4t^2y = e^{t},\qquad y(1)=0,\qquad t<0.
\end{equation*}

First divide by $t^3$ on both side of the equation, we get
$$y' + \frac{4}{t}y = \frac{e^{t}}{t^3}$$
Using the method of integrating factor we have equation for $u(t)$
$$u(t) = e^{\int \frac{4}{t}dt} = e^{4\ln(t) + c} = t^4$$
where constant $c$ is arbitrary, it's chosen to be 0 here. Then
$$\bigl(y u(t)\bigr)' = u(t)\frac{e^{t}}{t^3}$$
rearranging gives equation
$$y = \frac{1}{u(t)}\int u(t)\frac{e^{t}}{t^3}$$
substitute in $u(t) = t^4$
$$y = \frac{1}{t^4}\int te^{t}$$
use integration by parts
$$y = \frac{e^{t}}{t^3}  \frac{e^{t}}{t^4} + \frac{c_1}{t^4}$$
to check $c_1$, plug in condition $y(1) = 0$
$$y(1) = e  e + c_1 = c_1= 0$$
Plug in $c_1 = 0$ gets
$$y = \frac{e^{t}}{t^3}  \frac{e^{t}}{t^4} $$

My solution to this quiz can be found on this attachment.

Nick, still typed in forum solution is better then typed externally (but I applaud your typesetting skills)