Internal problem ID [1130]
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: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x +\left (x^{2}+2\right ) y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \sin \relax (x ) x \end {align*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 15
dsolve([x^2*diff(y(x),x$2)-2*x*diff(y(x),x)+(x^2+2)*y(x)=0,x*sin(x)],y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \relax (x ) x +c_{2} x \cos \relax (x ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 33
DSolve[x^2*y''[x]-2*x*y'[x]+(x^2+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 e^{-i x} x-\frac {1}{2} i c_2 e^{i x} x \\ \end{align*}