6.9.23 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 : Tuesday, September 30, 2025 at 05:19:16 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*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

ode:=x^2*diff(diff(y(x),x),x)-(2*a-1)*x*diff(y(x),x)+a^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 18

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

\begin{align*} y(x)&\to x^a (a c_2 \log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**2*y(x) + x**2*Derivative(y(x), (x, 2)) - x*(2*a - 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x^{\operatorname {re}{\left (a\right )}} \left (C_{3} \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (a\right )}}\right | \right )} + C_{4} \cos {\left (\log {\left (x \right )} \operatorname {im}{\left (a\right )} \right )} + \left (C_{1} \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (a\right )}}\right | \right )} + C_{2} \cos {\left (\log {\left (x \right )} \operatorname {im}{\left (a\right )} \right )}\right ) \log {\left (x \right )}\right ) \]