Question

Find a particular solution to y′′+8y′+16y=−2e^−4t.

Answer 1

i t is Undetermined Coefficients kind problem

Answer 2

1/ homogenous part give you \(y_c = C_1 e^{-4t}+ C_2t e^{-4t}\)
2/ therefore, partial solution must have the form of \(y_p=At^2e^{-4t}\)
3/ take first and second derivative of \(y_p\), then replace them into the original function to solve for A, you will have A = -1
replace to \(y_p\), you have \(y_p = -t^2e^{-4t}\)
4/ combine homogeneous and partial solution, your answer should be
\[y = C_1e^{-4t}+C_2te^{-4t}-t^2e^{-4t}\)

Answer 3

thank you very much! that helped a lot

Answer 4

ok