73.9.22 problem 14.2 (L)

Internal problem ID [15283]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.2 (L)
Date solved : Tuesday, January 28, 2025 at 07:51:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+\left (x^{2}+2\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x \sin \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.004 (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)],singsol=all)
 

\[ y = x \left (\sin \left (x \right ) c_{1} +\cos \left (x \right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 33

DSolve[x^2*D[y[x],{x,2}]-2*x*D[y[x],x]+(x^2+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 e^{-i x} x-\frac {1}{2} i c_2 e^{i x} x \]