# find the general solution(ode)

Isn't the general solution yh + yp? What am I doing wrong? Got everything else correct, but stuck on question d.

• Did you forgot $e^t$ (it should be $t^2 e^t$) – Arctic Char Oct 30 '19 at 2:35
• I entered $$c1e^t + c2te^t -2t + c1 -t^2e^t + c2$$ and it's still wrong – jimmy kudo Oct 30 '19 at 2:38
• What about that $2t$? where does it come from? – Arctic Char Oct 30 '19 at 2:39
• $$c1e^t + c2te^t -2te^t + c1 -t^2e^t + c2$$ is incorrect – jimmy kudo Oct 30 '19 at 2:40
• Ok its y1u1 + u2y2 – jimmy kudo Oct 30 '19 at 3:05

HINT

\begin{align*} y^{\prime\prime} - 2y^{\prime} + y = (y^{\prime} - y)^{\prime} - (y^{\prime} - y) \end{align*}
Then make the substitution $$u = y^{\prime} - y$$, which reduces the problem to solving a first order ODE twice.
Your $$y_h = c_1e^t+c_2te^t$$ is correct but the $$y_p$$ is not . To find $$y_p$$ they used the method of variation of parameters as explained in the solution but they have their $$y_p$$ incorrect. The correct $$y_p=-t^2e^t$$ does not need the $$c_1$$ and $$c_2$$ but it uses $$u_1e^t+u_2te^t$$ where $$u_1$$ and $$u_2$$ are found by the integrals.
The correct answer is $$y= c_1e^t+c_2te^t -t^2e^t$$