54.1.322 problem 328

Internal problem ID [11636]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 328
Date solved : Tuesday, September 30, 2025 at 09:51:10 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} a \,x^{2} y^{n} y^{\prime }-2 x y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 33

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

\[ y^{2 n} \left (y^{n} a x -n -2\right )^{n} x^{-n}-c_1 = 0 \]

✓ Mathematica. Time used: 0.134 (sec). Leaf size: 42

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

\[ \text {Solve}\left [\frac {n \left (\log (x)-\log \left (-a x y(x)^n+n+2\right )\right )}{n+2}-\frac {2 n \log (y(x))}{n+2}=c_1,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) + y(x)/(x*(a*x*y(x)**n - 2)) cannot be solve