How Do You Solve a Tricky Second Order Nonhomogeneous Differential Equation?

[whatababy]

Homework Statement

solve the initial value problem:
y^''+8.4y^'+17.64y=e^-4.2x
y(0)=1, y^'(0)=1

y(x)=?

Homework Equations

y=y_c+y_p

The Attempt at a Solution

The way I tried to achieve is to solve the corresponding homo equation first:
y^''+8.4y^'+17.64y=0, which gives y_c;
y_c=c₁e^-4.2x+c₂e^-4.2xx

Then try to find y_p, generally I would assume a y_p=Ae^-4.2x, but from the y_c got above, clearly y_p=Ae^-4.2x or y_p=Ae^-4.2xx is not good. If I add one more 'x' in y_p assumption,in which it seems A has to be 0, which is not right either...

Any idea? Or I made some mistake?

Meric.

 

[Mark44]

Assuming that the work you show is correct, y_p = Ax²e^-4.2x is the right choice. Plug it into your DE and solve for A.