Differential Equations | Part IA, 2004

Find two linearly independent solutions of the differential equation

d2ydx2+2pdydx+p2y=0\frac{d^{2} y}{d x^{2}}+2 p \frac{d y}{d x}+p^{2} y=0

Find also the solution of

d2ydx2+2pdydx+p2y=epx\frac{d^{2} y}{d x^{2}}+2 p \frac{d y}{d x}+p^{2} y=e^{-p x}

that satisfies

y=0,dydx=0 at x=0y=0, \frac{d y}{d x}=0 \quad \text { at } x=0

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