76.39.13 problem Ex. 13

Internal problem ID [20230]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VII. Linear equations with variable coefficients. End of chapter problems at page 91
Problem number : Ex. 13
Date solved : Thursday, October 02, 2025 at 05:35:01 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 35

ode:=x^2*diff(diff(y(x),x),x)-(2*m-1)*x*diff(y(x),x)+(m^2+n^2)*y(x) = n^2*x^m*ln(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (c_2 n -1\right ) \sin \left (n \ln \left (x \right )\right )+n \left (c_1 \cos \left (n \ln \left (x \right )\right )+\ln \left (x \right )\right )\right ) x^{m}}{n} \]

✓ Mathematica. Time used: 0.232 (sec). Leaf size: 79

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

\begin{align*} y(x)&\to x^m \left (c_1 x^{-\frac {n \sqrt {m^2+n^2}}{\sqrt {-m^2-n^2}}}+c_2 x^{\frac {n \sqrt {m^2+n^2}}{\sqrt {-m^2-n^2}}}+\log (x)\right ) \end{align*}

✓ Sympy. Time used: 139.782 (sec). Leaf size: 393

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

\[ \text {Solution too large to show} \]