54.2.172 problem 749

Internal problem ID [12046]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 749
Date solved : Sunday, October 12, 2025 at 02:13:59 AM
CAS classification : [_rational]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 152

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

\begin{align*} y &= \frac {\sqrt {\left (\left (x^{2}-1\right ) {\mathrm e}^{2 x^{2}}+c_1 \,{\mathrm e}^{-\frac {1}{2}} \left (x^{2}+1\right )\right ) \left ({\mathrm e}^{2 x^{2}}+c_1 \,{\mathrm e}^{-\frac {1}{2}}\right ) {\mathrm e}^{-2 x^{2}} {\mathrm e}^{-x^{4}}}\, {\mathrm e}^{\frac {x^{4}}{2}}}{c_1 \,{\mathrm e}^{-\frac {1}{2}-x^{2}}+{\mathrm e}^{x^{2}}} \\ y &= -\frac {\sqrt {\left (\left (x^{2}-1\right ) {\mathrm e}^{2 x^{2}}+c_1 \,{\mathrm e}^{-\frac {1}{2}} \left (x^{2}+1\right )\right ) \left ({\mathrm e}^{2 x^{2}}+c_1 \,{\mathrm e}^{-\frac {1}{2}}\right ) {\mathrm e}^{-2 x^{2}} {\mathrm e}^{-x^{4}}}\, {\mathrm e}^{\frac {x^{4}}{2}}}{c_1 \,{\mathrm e}^{-\frac {1}{2}-x^{2}}+{\mathrm e}^{x^{2}}} \\ \end{align*}

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 155

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

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

✗ Sympy

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

NotImplementedError : The given ODE -x**5/y(x) + 2*x**3*y(x) - x*y(x)**3 + Derivative(y(x), x) cannot be solved by the factorable group method