<s>[INST] Finding solution to differential equation I'm stuck on this problem:
Find the solution of the given initial value problem:
$ty′+4y=t^2−t+5$, $y(1)=7$, $t>0$
When I multiply both sides by $\mu(t)$ I find that $\mu(t) = e^{4t}$
So:
$$
\frac{d}{dt} (\mu y) =  \mu t^2 - \mu t + 5 \mu\\
e^{4t} y= \int t^2 e^{4t}-te^{4t}+5e^{4t} dt
$$
This is what I have so far
 [/INST] For $t y′ + 4y = t^2 − t + 5$ then
\begin{align}
\frac{1}{\mu} \frac{d}{dt} \, ( \mu \, y) &= y' + \frac{\mu'}{\mu} \, y
\end{align}
which leads to 
$$ \frac{d}{dt} \ln \mu = \frac{4}{t} = \frac{d}{dt} (4 \ln t) = \frac{d}{dt} \ln(t^4),$$
or $\mu(t) = t^4$. Now,
\begin{align}
y' + \frac{4}{t} \, y &= t - 1 + \frac{5}{t} \\
\frac{1}{t^4} \, \frac{d}{dt} \, (t^4 \, y) &= t - 1 + \frac{5}{t} \\
\frac{d}{dt} \, (t^4 \, y) &= t^5 - t^4 + 5 \, t^3 \\
y(t) &= t^{-4} \, \int^{t} (u^{5} - u^{4} + 5 \, u^{3}) \, du \\
&= \frac{t^2}{6} - \frac{t}{5} + \frac{5}{4} + \frac{c_{0}}{t^4}.
\end{align}
By applying $y(1) = 7$ leads to 
$$y(t) = \frac{10 t^6 - 12 t^5 + 75 t^4 + 347}{60 \, t^4}.$$
</s>