Please find y.?

Question

Please find y.

y”+ a y’ = k exp (-bx);

y” stands for d^2y /dx^2.

a, k and b are constants.

Given that
When x = 0, y = 0 and y’ =0

Answer 1

The caracteristic equation is
r(r+a)=0 so r= 0 and r=-a
The homogeneos equation has solution y=C-1+C-_2 *e^-ax
A particular solution with second side could be
m(exp-bx)=y
y´=-bm*exp(-bx) and y´´ = b^2m*exp(-bx)
so exp(-bx)[b^2m-abm]=k exp(-bx)
and m=k/(b^2-ab) and the solution is
y=C_1+C-2*e^-ax+ k/(b^2-ab) *exp(-bx)
x=0
y=C_1+C_2+k/(b^2-ab)=0
y´=-aC_2 e^-ax -k/(b-a) exp(-bx)
x=0
-aC_2-k/(b-a)=0 so C_2 = - k/(ab-a^2). You can find C_1