54.2.161 problem 738

Internal problem ID [12035]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 738
Date solved : Wednesday, October 01, 2025 at 12:01:23 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\frac {2 a}{-x^{2} y+2 a y^{4} x^{2}-16 a^{2} x y^{2}+32 a^{3}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 964

ode:=diff(y(x),x) = 2*a/(-x^2*y(x)+2*a*y(x)^4*x^2-16*a^2*x*y(x)^2+32*a^3); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.284 (sec). Leaf size: 335

ode=D[y[x],x] == (2*a)/(32*a^3 - x^2*y[x] - 16*a^2*x*y[x]^2 + 2*a*x^2*y[x]^4); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x\left (\frac {4 a y(x)^3+4 a y(x)^2+1}{-16 y(x) a^2-16 a^2+K[1] \left (4 a y(x)^3+4 a y(x)^2+1\right )}-\frac {y(x)^2}{K[1] y(x)^2-4 a}\right )dK[1]+\int _1^{y(x)}\left (-\frac {2 x K[2]}{x K[2]^2-4 a}-\int _1^x\left (\frac {2 K[1] K[2]^3}{\left (K[1] K[2]^2-4 a\right )^2}-\frac {2 K[2]}{K[1] K[2]^2-4 a}+\frac {12 a K[2]^2+8 a K[2]}{-16 K[2] a^2-16 a^2+K[1] \left (4 a K[2]^3+4 a K[2]^2+1\right )}-\frac {\left (4 a K[2]^3+4 a K[2]^2+1\right ) \left (K[1] \left (12 a K[2]^2+8 a K[2]\right )-16 a^2\right )}{\left (-16 K[2] a^2-16 a^2+K[1] \left (4 a K[2]^3+4 a K[2]^2+1\right )\right )^2}\right )dK[1]-\frac {4 \left (4 a^2-3 x K[2]^2 a-2 x K[2] a\right )}{4 a x K[2]^3+4 a x K[2]^2-16 a^2 K[2]-16 a^2+x}\right )dK[2]=c_1,y(x)\right ] \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-2*a/(32*a**3 - 16*a**2*x*y(x)**2 + 2*a*x**2*y(x)**4 - x**2*y(x)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -2*a/(32*a**3 - 16*a**2*x*y(x)**2 + 2*a*x**2*y(x)**4 - x**2*y(x)) + Derivative(y(x), x) cannot be solved by the lie group method