Consider the boundary-value problem $$\varepsilon y''+x(x+3)^2y=0$$ for $a<x<b$
Find a first term WKB expansion of the solution in cases where, $a=0,y(a)=0,b=1,y(b)=1$
I have constructed the solution in the following form $$y\sim \frac1{\left( x \left( x+3 \right)^2 \right)^\frac14} \left( 2a_R\cos\left( \frac1{\varepsilon} \left( \frac25 x^\frac52 + 2x^\frac32 \right)-\frac{\pi}4\right) + b_R\cos\left( \frac1{\varepsilon} \left( \frac25 x^\frac52 + 2x^\frac32 \right)+\frac{\pi}4\right)\right) $$ There are two unknown constants in this equation as $a_R,b_R$. I can substitute $y(1)=1$ into the above equation to get one equation, but the boundary condition for $y(0)=0$ does not make sense since the solution is not defined at $0$ How do I deal with this case?