I have been given this task as a assignment but I this far not able to come anywhere near a solution.
Let $p_1,p_2$ continous gunctions from $\mathbb{R}$ to $\mathbb{R}$. The following Differential equation $$y''+p_1(x)y'+p_2(x)y=0 \ \ \ \ \ (*)$$ has as a solution $y_1(x)=\sqrt{1+x^2}$ and the Wronskian of every 2 solutions of (*) is constant.
Find the basis solutions for (*) and the functions $p_1$ and $p_2$.
I tried using that the derivative of the Wronskian would be 0 but it didn't leed to anything.
Can anyone give me any hints into finding the solution for both tasks. Thank you very much in advance.
$W$ is constant so: $$y'_2y_1-y_2y'_1=C$$ $$ \left( \dfrac {y_2}{y_1} \right)'=\dfrac C {y_1^2}$$ This is a first order Differential equation where $y_1$ is given. $$y_1= \sqrt {1+x^2}$$ Then you will have two equations to solve in $p_1,p_2$: $$y_2''+p_1(x)y_2'+p_2(x)y_2=0$$ $$y_1''+p_1(x)y_1'+p_2(x)y_1=0$$