2.II.7A

Find the first three non-zero terms in series solutions $y_{1}(x)$ and $y_{2}(x)$ for the differential equation

$x \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+4 x^{3} y=0$

that satisfy the boundary conditions

$\begin{aligned} &y_{1}(0)=a, \quad y_{1}^{\prime \prime}(0)=0, \\ &y_{2}(0)=0, \quad y_{2}^{\prime \prime}(0)=b, \end{aligned}$

where $a$ and $b$ are constants.

Determine the value of $\alpha$ such that the change of variable $u=x^{\alpha}$ transforms $(*)$ into a differential equation with constant coefficients. Hence find the general solution of $(*)$.

*Typos? Please submit corrections to this page on GitHub.*