# Math Help - Finding general solution

1. ## Finding general solution

Given that three solutions to the inhomongenous DE equation y''+p(t)y'+q(t)y=g(t)

are y1=1+e^(t^2), y2= 1+te^(t^2), y3= 1+(t+1)e^(t^2),

find the general solution

2. Since $y_1$, $y_2$ and $y_3$ all satisfy the ODE then

$u_1 = y_3 - y_2 = e^{t^2}$ and $u_2 = y_3 - y_2 = t e^{t^2}$

satisfy

$u'' + pu' + q u = 0$

Thus, the general solution is

$y = c_1 e^{t^2} + c_2 t e^{t^2} + 1$.

Note that we could have used either $y_1, y_2\; \text{or}\; y_3$ for $y_p$ but can adjust the constants such that all we need is $y_p =1$.

3. Why did you subtract the solutions? Thanks

4. It was to get to the homogeneous ODE

If $y_1$ and $y_2$ satisfy your ODE, i.e.

$y_1'' + py_1' + q y_1 = g$

and

$y_2'' + py_2' + q y_2 = g$

then substracting gives

$(y_2''-y_1'') + p(y_2'-y_1') + q(y_2-y_1) = 0$.