2.I.2B

Differential Equations
Part IA, 2001

The function y(x)y(x) satisfies the inhomogeneous second-order linear differential equation

yy2y=18xex.y^{\prime \prime}-y^{\prime}-2 y=18 x e^{-x} .

Find the solution that satisfies the conditions that y(0)=1y(0)=1 and y(x)y(x) is bounded as xx \rightarrow \infty.