Hi guys! I ve got this very difficult exercise for my uni. Could someone please help? Thank you in advance.
You should end up with
$\displaystyle p''(t) = a[d_0 + d_1p(t) - (s_0 + s_1p(t))]$
$\displaystyle = ad_0 + ad_1p(t) - as_0 - as_1p(t)$
$\displaystyle = (ad_1 - as_1)p(t) + ad_0 - as_0$.
So $\displaystyle p''(t) + (as_1 - ad_1)p(t) = ad_0 - as_0$.
Solve the homogeneous characteristic equation:
$\displaystyle m^2 + as_1 - ad_1 = 0$
$\displaystyle m^2 = ad_1 - as_1$
$\displaystyle m^2 = a(d_1 - s_1)$.
Now since $\displaystyle d_1 < 0$ and $\displaystyle s_1 > 0$, this means $\displaystyle d_1 - s_1 < 0$.
Also, since $\displaystyle a > 0$, this means $\displaystyle a(d_1 - s_1) < 0$.
So you have $\displaystyle m^2$ equaling a negative number. What does this tell you about $\displaystyle m$? Can you solve the DE now?