11.3 problem Problem 3

Internal problem ID [2305]

Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section: Chapter 8, Linear differential equations of order n. Section 8.9, Reduction of Order. page 572
Problem number: Problem 3.
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 &= x \sin \relax (x ) \end {align*}

Solution by Maple

Time used: 0.015 (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: 0.012 (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*}