I have the BVP:
$$ \frac{d^2y}{dx^2}=e^{2x}+y\frac{dy}{dx} $$
With the following mixed boundary conditions: $y(0)=1$ and $y'(1)=0$
How would I apply the shooting method to this BVP? It's throwing me off that the right hand BC is a derivative, so I'm not sure how to approach this.
Your equation is second order, so you can use a state vector $u=[u_0,u_1]=[y,y']$ giving a first order system $$ u'=F(x,u)=[u_1,\;\exp(2x)+u_0u_1]. $$ With a numerical solution of this system you can build a shooting function $S$ taking as input a scalar $v$ for an initial state $u(0)=[1,v]$ and returning $S(v)=u_1(1)$, the second component of the state at the end of the integration interval.
Now apply your preferred root-finder to the scalar function $S$.
