12.9.23 problem 23

Internal problem ID [1779]
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 : 23
Date solved : Monday, January 27, 2025 at 05:34:45 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }-\left (2 a -1\right ) x y^{\prime }+a^{2} y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{a} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 14

dsolve([x^2*diff(y(x),x$2)-(2*a-1)*x*diff(y(x),x)+a^2*y(x)=0,x^a],singsol=all)
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) x^{a} \]

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 18

DSolve[x^2*D[y[x],{x,2}]-(2*a-1)*x*D[y[x],x]+a^2*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x^a (a c_2 \log (x)+c_1) \]