12.9.24 problem 24

Internal problem ID [1780]
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
Date solved : Monday, January 27, 2025 at 05:34:46 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: 0.005 (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)
 

\[ y = x \left (c_1 \sin \left (x \right )+c_2 \cos \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.027 (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]
 

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