Differential Equations | Part IA, 2008

Find the first three non-zero terms in series solutions y1(x)y_{1}(x) and y2(x)y_{2}(x) for the differential equation

xd2ydx2dydx+4x3y=0x \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+4 x^{3} y=0

that satisfy the boundary conditions

y1(0)=a,y1(0)=0,y2(0)=0,y2(0)=b,\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 aa and bb are constants.

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

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