Here's the problem:
$\frac{dy}{dt}-y=7e^t + 25e^{6t}$ in terms of $\,y\,$, when $\,y(0)=7\,$
Here's the problem:
$\frac{dy}{dt}-y=7e^t + 25e^{6t}$ in terms of $\,y\,$, when $\,y(0)=7\,$
To solve:
$y'-y=7e^t+25e^{6t}$
Multiply by $e^{-t}$:
$e^{-t}y'-e^{-t}y=7+25e^{5t}$
So that the right side is the result of the product rule, as follows:
$(ye^{-t})'=7+25e^{5t}$
Integrate both sides to find
$ye^{-t}=7t+5e^{5t}+C$ $y=7te^t+5e^{6t}+Ce^t$
Then just plug in the point $(t,y)=(0,7)$ given by your initial condition and solve for $C$. The resulting function for $y$ will be your answer. Though as a disclaimer I do have to point out that unless you have mathematically minded friends, this will probably impress them less than you'd like.
Put $y = a e^t + b te^t + c e^{6t}$ and solve for $a$, $b$, $c$.