Differential Equations | Part IA, 2004

Construct a series solution y=y1(x)y=y_{1}(x) valid in the neighbourhood of x=0x=0, for the differential equation

d2ydx2+4x3dydx+x2y=0\frac{d^{2} y}{d x^{2}}+4 x^{3} \frac{d y}{d x}+x^{2} y=0


y1=1,dy1dx=0 at x=0.y_{1}=1, \frac{d y_{1}}{d x}=0 \quad \text { at } x=0 .

Find also a second solution y=y2(x)y=y_{2}(x) which satisfies

y2=0,dy2dx=1 at x=0.y_{2}=0, \frac{d y_{2}}{d x}=1 \quad \text { at } x=0 .

Obtain an expression for the Wronskian of these two solutions and show that

y2(x)=y1(x)0xeξ4y12(ξ)dξy_{2}(x)=y_{1}(x) \int_{0}^{x} \frac{e^{-\xi^{4}}}{y_{1}^{2}(\xi)} d \xi

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