60.1.322 problem 328

Internal problem ID [10336]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 328
Date solved : Sunday, March 30, 2025 at 04:12:23 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: 0.074 (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);
 

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

✓ Mathematica. Time used: 0.554 (sec). Leaf size: 170

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]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {n \left (a x K[2]^n-2\right )}{K[2] \left (-a x K[2]^n+n+2\right )}-\int _1^x\left (\frac {a^2 n^2 K[1] K[2]^{2 n-1}}{(n+2) \left (a K[1] K[2]^n-n-2\right )^2}-\frac {a n^2 K[2]^{n-1}}{(n+2) \left (a K[1] K[2]^n-n-2\right )}\right )dK[1]\right )dK[2]+\int _1^x\left (\frac {n}{(n+2) K[1]}-\frac {a n y(x)^n}{(n+2) \left (a K[1] y(x)^n-n-2\right )}\right )dK[1]=c_1,y(x)\right ] \]

✗ 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 solved by the factorable group method