I need help finding the general solution to the differential equation $y''(t)+7y'(t)=-14$ What I've tried:
I have the inhomogeneous differential equation:
$$y''(t)+7y'(t)=-14$$
I find the particular solution to be on the form $$kt$$
by inserting the particular solution in the equation
$$(kt)''+7(kt)'=-14$$
and isolating for k, I get that:
$$k=-2$$
and therefore the particular solution is
$$y(t)-2t$$
I also need the general solution for the homogenous equation
$$y''(t)+7y'(t)=0$$
by finding the roots of the characteristic polynomial
$$z^2+7z=z(z+7)=0$$
$$z_1=0$$
$$z_2=-7$$
I get the general solution:
$$c_1e^{0t}+c_2e^{-7t}=c_1+c_2e^{-7t}$$
Now, according to my textbook, the general solution of an inhomogeneous differential equation is given by
$$y(t)=y_p(t)+y_{hom}(t)$$
Where $y_p(t)$ is the particular solution and $y_{hom}(t)$ is the general solution to the homogenous equation. Therefore I get the general solution to be
$$y(t)=c_1+c_2e^{-7t}-2t$$
This is not consistent with Maple's result however

Can anyone see where I've gone wrong?

You went wrong when you thought what Maple wrote is different from your solution in any significant way.
Maple has swapped the roles of $c_1$ and $c_2$ compared to you. And Maple's $c_1$ is seven times larger than your $c_2$ and has opposite sign, but since the constants are arbitrary anyways this doesn't matter. So you and Maple describe the exact same collection of functions.
