23.1.152 problem 153

Internal problem ID [4759]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 153
Date solved : Tuesday, September 30, 2025 at 08:30:41 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 34

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

\[ y = \frac {\left (-1+\left (n +1\right ) \ln \left (x \right )\right ) x \,x^{n}+c_1 \left (n +1\right )^{2}}{\left (n +1\right )^{2} x} \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 29

ode=x*D[y[x],x]==x^n*Log[x]-y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^n ((n+1) \log (x)-1)}{(n+1)^2}+\frac {c_1}{x} \end{align*}

✓ Sympy. Time used: 0.205 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = \frac {C_{1}}{x} - \frac {e^{n \log {\left (x \right )}}}{n^{2} + 2 n + 1} + \frac {e^{n \log {\left (x \right )}} \log {\left (x \right )}}{n + 1} \]