I am trying to solve $$ y''' - y = -ye^{2t}. $$ with initial values $y(0) = 0$, $y'(0) = -1$, $y''(0) = -3$
The problem I am running in to is when I am transforming the left and right side of the equation: $$ L{y'''} - L{y} = - L{ye^{2t}} $$ Using rules for transformation I get: $$ Y(s)s^3 + 3s^2 +s - Y(s) = Y(s-2). $$ Normally one would just solve for Y and do the inverse Laplace transform. However, now the right hand side is S-shifted.
Is there a general method for solving this, or any tricks one might apply to get rid of the S-shift or $e^{2t}$ term from the original equation?