9.11 problem 11

Internal problem ID [1117]

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: 11.
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 }-x \left (2 x -1\right ) y^{\prime }+\left (x^{2}-x -1\right ) y-{\mathrm e}^{x} x^{2}=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{x} x \end {align*}

Solution by Maple

Time used: 0.013 (sec). Leaf size: 24

dsolve([x^2*diff(y(x),x$2)-x*(2*x-1)*diff(y(x),x)+(x^2-x-1)*y(x)=x^2*exp(x),x*exp(x)],y(x), singsol=all)
 

\[ y \relax (x ) = \frac {{\mathrm e}^{x} c_{2}}{x}+x \,{\mathrm e}^{x} c_{1}+\frac {x^{2} {\mathrm e}^{x}}{3} \]

Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 32

DSolve[x^2*y''[x]-x*(2*x-1)*y'[x]+(x^2-x-1)*y[x]==x^2*Exp[x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {e^x \left (2 x^3+3 c_2 x^2+6 c_1\right )}{6 x} \\ \end{align*}