54.1.496 problem 509

Internal problem ID [11810]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 509
Date solved : Sunday, October 12, 2025 at 01:58:47 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

\begin{align*} 9 y^{4} \left (x^{2}-1\right ) {y^{\prime }}^{2}-6 x y^{5} y^{\prime }-4 x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.439 (sec). Leaf size: 225

ode:=9*y(x)^4*(x^2-1)*diff(y(x),x)^2-6*x*y(x)^5*diff(y(x),x)-4*x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}} \\ y &= -2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\ y &= \frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\ y &= -\frac {\left (i \sqrt {3}-1\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\ y &= \frac {\left (1+i \sqrt {3}\right ) 2^{{1}/{3}} \left (-x^{2}+1\right )^{{1}/{6}}}{2} \\ y &= \frac {2^{{2}/{3}} {\left (\left (-4 c_1^{2}+x^{2}-1\right ) c_1^{2}\right )}^{{1}/{3}}}{2 c_1} \\ y &= -\frac {2^{{2}/{3}} {\left (\left (-4 c_1^{2}+x^{2}-1\right ) c_1^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 c_1} \\ y &= \frac {2^{{2}/{3}} {\left (\left (-4 c_1^{2}+x^{2}-1\right ) c_1^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.267 (sec). Leaf size: 199

ode=-4*x^2 - 6*x*y[x]^5*D[y[x],x] + 9*(-1 + x^2)*y[x]^4*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt [3]{-\frac {1}{2}} \sqrt [3]{-4 x^2+4+c_1{}^2}}{\sqrt [3]{c_1}}\\ y(x)&\to -1\\ y(x)&\to 0\\ y(x)&\to \sqrt [3]{-\frac {1}{2}}\\ y(x)&\to \text {Indeterminate}\\ y(x)&\to -\sqrt [3]{-2} \sqrt [6]{1-x^2}\\ y(x)&\to \sqrt [3]{-2} \sqrt [6]{1-x^2}\\ y(x)&\to -\sqrt [3]{2} \sqrt [6]{1-x^2}\\ y(x)&\to \sqrt [3]{2} \sqrt [6]{1-x^2}\\ y(x)&\to -(-1)^{2/3} \sqrt [3]{2} \sqrt [6]{1-x^2}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{2} \sqrt [6]{1-x^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**2 - 6*x*y(x)**5*Derivative(y(x), x) + (9*x**2 - 9)*y(x)**4*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out