Paper 2, Section II, 7C7 \mathrm{C}

Differential Equations | Part IA, 2009

Consider the differential equation

xd2y dx2+(cx)dy dxy=0,x \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+(c-x) \frac{\mathrm{d} y}{\mathrm{~d} x}-y=0,

where cc is a constant with 0<c<10<c<1. Determine two linearly independent series solutions about x=0x=0, giving an explicit expression for the coefficient of the general term in each series.

Determine the solution of

xd2y dx2+(cx)dy dxy=xx \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+(c-x) \frac{\mathrm{d} y}{\mathrm{~d} x}-y=x

for which y(0)=0y(0)=0 and dy/dx\mathrm{d} y / \mathrm{d} x is finite at x=0x=0.

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