14.10.8 problem Problem 21

Internal problem ID [3780]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.8, A Differential Equation with Nonconstant Coefficients. page 567
Problem number : Problem 21
Date solved : Tuesday, September 30, 2025 at 06:57:18 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 33

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

\[ y = x^{m} \left (c_2 +\ln \left (x \right ) c_1 +\frac {\ln \left (x \right )^{k} \ln \left (x \right )^{2}}{k^{2}+3 k +2}\right ) \]

✓ Mathematica. Time used: 0.039 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to x^m \left (\frac {\log ^{k+2}(x)}{k^2+3 k+2}+c_2 m \log (x)+c_1\right ) \end{align*}

✓ Sympy. Time used: 2.233 (sec). Leaf size: 121

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

\[ y{\left (x \right )} = x^{\operatorname {re}{\left (m\right )}} \left (C_{3} \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )} + C_{4} \cos {\left (\log {\left (x \right )} \operatorname {im}{\left (m\right )} \right )} + \left (C_{1} \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )} + C_{2} \cos {\left (\log {\left (x \right )} \operatorname {im}{\left (m\right )} \right )}\right ) \log {\left (x \right )} + \left (\log {\left (x \right )} \int \frac {x^{m - \operatorname {re}{\left (m\right )} - 1} \log {\left (x \right )}^{k}}{\sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )}}\, dx - \int \frac {x^{m - \operatorname {re}{\left (m\right )} - 1} \log {\left (x \right )}^{k + 1}}{\sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )}}\, dx\right ) \sin {\left (\log {\left (x \right )} \left |{\operatorname {im}{\left (m\right )}}\right | \right )}\right ) \]