Q. Consider the ODE: $u''(t)+P(t)u'(t)+Q(t)u(t)=R(t),\ t\in[0,\ 1]$. There exists continuous functions $P,Q$ and $R$ defined on $[0,\ 1]$ and two solutions $u_1$ and $u_2$ of this ODE such that
- $W(t)=2t-1 \ \ \forall t\in[0,\ 1]$
- $W(t)=\sin 2\pi t \ \ \forall t\in[0, \ 1]$
- $W(t)=\cos 2\pi t\ \ \forall t\in[0,\ 1]$
- $W(t)=1 \ \ \forall t\in[0,\ 1]$
I observe that for some non-homogeneous problems like $$y''-3y'+2y=2,$$the wronskian $W(t)=W(y_1(t),y _2(t))$ for the linearly independent solutions $y_1(t)=e^t+1$ and $y_2(t)=e^{2t}+1$ do have an isolated zero on $\mathbb R$. How can we find $P,Q$ and $R$ satisfying $W(t)=2t-1$ for some solutions $y_1$ and $y_2$ or how to reject the possibility?