75.12.28 problem 302

Internal problem ID [16823]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 12. Miscellaneous problems. Exercises page 93
Problem number : 302
Date solved : Monday, March 31, 2025 at 03:23:12 PM
CAS classification : [[_homogeneous, `class G`], _rational]

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

✓ Maple. Time used: 0.144 (sec). Leaf size: 32

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

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

✓ Mathematica. Time used: 0.533 (sec). Leaf size: 160

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
n = symbols("n") 
y = Function("y") 
ode = Eq(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*(x*y(x)**n - 2)) cannot be solved by the factorable group method