Internal problem ID [1121]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page
253
Problem number: 15.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-\left (1+2 x \right ) y^{\prime }+\left (x +1\right ) y+{\mathrm e}^{-x}=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} \end {align*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
dsolve([x*diff(y(x),x$2)-(2*x+1)*diff(y(x),x)+(x+1)*y(x)=-exp(-x),exp(x)],y(x), singsol=all)
\[ y \relax (x ) = c_{2} {\mathrm e}^{x}+{\mathrm e}^{x} c_{1} x^{2}+\frac {\left (2 x -1\right ) {\mathrm e}^{-x}}{4}-\expIntegral \left (1, 2 x \right ) x^{2} {\mathrm e}^{x} \]
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 46
DSolve[x*y''[x]-(2*x+1)*y'[x]+(x+1)*y[x]==-Exp[-x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} e^{-x} \left (2 e^{2 x} \left (2 x^2 \text {Ei}(-2 x)+c_2 x^2+2 c_1\right )+2 x-1\right ) \\ \end{align*}