Consider ODE $\dot x(t) = λx(t) + cos(t)e^{λt}, t > 0$, $x(0)=x_0,\lambda \in \mathbb{R}$. Investigate existence and uniqueness of solutions with respect to λ and find a closed representation of the solution $x(t)$.
I have just started learning the existence and uniqueness of solution of ODE and I found this exercise from my textbook. However, I don't know which theorem I should use to investigate the existence and uniqueness of above ODE. Can anyone give me some hints?
For approximate of solution part, I guess if $\lambda <0$ then $x'\approx \lambda x(t)$ and $\dot x \approx cos(t)e^{\lambda t}, \lambda >0$. Am I corrected? Thank you in advance
If you set $$f(t,y)=\lambda y+\cos(t)e^{\lambda t},$$ this function is Lipschitz refer to $y$, and thus, by Picard theorem $$\dot x(t)=f(t,x(t))$$ has a unique solution.