Internal problem ID [1116]
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: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+2 x \left (x -1\right ) y^{\prime }+\left (x^{2}-2 x +2\right ) y-{\mathrm e}^{2 x} x^{3}=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= x \,{\mathrm e}^{-x} \end {align*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 28
dsolve([x^2*diff(y(x),x$2)+2*x*(x-1)*diff(y(x),x)+(x^2-2*x+2)*y(x)=x^3*exp(2*x),x*exp(-x)],y(x), singsol=all)
\[ y \relax (x ) = x \,{\mathrm e}^{-x} c_{2}+x^{2} {\mathrm e}^{-x} c_{1}+\frac {{\mathrm e}^{2 x} x}{9} \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 30
DSolve[x^2*y''[x]+2*x*(x-1)*y'[x]+(x^2-2*x+2)*y[x]==x^3*Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{9} e^{-x} x \left (e^{3 x}+9 (c_2 x+c_1)\right ) \\ \end{align*}