$\displaystyle d^2y/dx^2 - dy/dx - dy= 3e^-x + 10sinx -4x$
please check if my answer are correct
$\displaystyle y=c1e^-x + c2e^2x -xe^-x+ cosx +3sinx+2x$
I think we may have a problem.
$\displaystyle \frac{d^2y}{dx}-\frac{dy}{dx}-y=3e^{-x}+10\sin(x)-4x$
solving the homogenious equation for the particular solution we get..
$\displaystyle \frac{d^2y}{dx}-\frac{dy}{dx}-y=0$
$\displaystyle m^2-m-1=0 \iff m^2-m+\frac{1}{4}=1+\frac{1}{4} \iff (x-\frac{1}{2})^2=\frac{5}{4} \iff x=\frac{1 \pm \sqrt{5}}{2}$
so
$\displaystyle y_c=c_1e^{\frac{1+\sqrt{5}}{2}x}+c_2e^{\frac{1-\sqrt{5}}{2}x}$
See what you can do from here.
then the equation is
$\displaystyle m^2-m-2=0 \iff (m-2)(m+1)$
so
$\displaystyle y_c=c_1e^{2x}+c_2e^{-x}$
Now we need to find the particular solution
since $\displaystyle e^{-x}$ in the complimentry solution
the particular solution will be of the form
$\displaystyle y_p=\underbrace{Ax^2+Bx+C}_{forThe-4x}+\underbrace{E\cos(x)+F\sin(x)}_{forThe10\sin(x )}+\underbrace{G(e^{-x})+H(xe^{-x})}_{forTheRepeted 3 e^{-x}}$