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 $a$, $b$, $c$.