54.2.146 problem 723

Internal problem ID [12020]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 723
Date solved : Tuesday, September 30, 2025 at 11:56:57 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 692

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

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

✓ Mathematica. Time used: 16.099 (sec). Leaf size: 672

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

\begin{align*} y(x)&\to -\frac {\sqrt [3]{-1024 a^6 c_1{}^3+9216 a^5 c_1 x-432 a^2+16 \sqrt {a^4 \left (\left (64 a^4 c_1{}^3-576 a^3 c_1 x+27\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{12 \sqrt [3]{2} a}-\frac {8 a^2 \left (3 x+a c_1{}^2\right )}{3 \sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {3} \sqrt {-a^4 \left (4096 a^7 c_1{}^4 x-8192 a^6 c_1{}^2 x^2+4096 a^5 x^3-128 a^4 c_1{}^3+1152 a^3 c_1 x-27\right )}}}+\frac {2 a c_1}{3}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1024 a^6 c_1{}^3+9216 a^5 c_1 x-432 a^2+16 \sqrt {a^4 \left (\left (64 a^4 c_1{}^3-576 a^3 c_1 x+27\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{24 \sqrt [3]{2} a}+\frac {4 \left (1+i \sqrt {3}\right ) a^2 \left (3 x+a c_1{}^2\right )}{3 \sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {3} \sqrt {-a^4 \left (4096 a^7 c_1{}^4 x-8192 a^6 c_1{}^2 x^2+4096 a^5 x^3-128 a^4 c_1{}^3+1152 a^3 c_1 x-27\right )}}}+\frac {2 a c_1}{3}\\ y(x)&\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1024 a^6 c_1{}^3+9216 a^5 c_1 x-432 a^2+16 \sqrt {a^4 \left (\left (64 a^4 c_1{}^3-576 a^3 c_1 x+27\right ){}^2-4096 a^5 \left (3 x+a c_1{}^2\right ){}^3\right )}}}{24 \sqrt [3]{2} a}+\frac {4 \left (1-i \sqrt {3}\right ) a^2 \left (3 x+a c_1{}^2\right )}{3 \sqrt [3]{-64 a^6 c_1{}^3+576 a^5 c_1 x-27 a^2+3 \sqrt {3} \sqrt {-a^4 \left (4096 a^7 c_1{}^4 x-8192 a^6 c_1{}^2 x^2+4096 a^5 x^3-128 a^4 c_1{}^3+1152 a^3 c_1 x-27\right )}}}+\frac {2 a c_1}{3} \end{align*}

✗ Sympy

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

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