Find the constants for IVP

Question

find the constant $a,b,c,y_0$ and $y'_0$ for IVP $ay''+by'+c=0$ with $y(0)=y_0$ and $y(0)=y'_0$ such that the solution decreases initially at some point later time $t>0$ the solution has local minimum and as $t$ and $y$ go to infinite. I really got stuck how to do it. This is not homework problems. Any help will be helpful.

Answer 1

You have to exclude oscillating solutions, thus $a\lambda^2+bλ+c=0$ can only have real solutions, which is the case for $b^2-4ac\ge 0$. Then your solution is $$ y(t)=c_1e^{λ_1(t-t_*)}+c_2e^{λ_2(t-t_*)} $$ where $t_*$ is the location of the minimum. There you want $$ 0=y'(t_*)=c_1λ_1+c_2λ_2\\ 0<y''(t_*)=c_1λ_1^2+c_2λ_2^2\\ ~------------~\\ 0<c_1λ_1(λ_1-λ_2) $$ Now play around with the sign variants of that last expression, sort the exponents so that $λ_1>λ_2$ to reduce the number of cases,...