60.1.386 problem 397

Internal problem ID [10400]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 397
Date solved : Sunday, March 30, 2025 at 04:36:54 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} {y^{\prime }}^{2}-2 x^{3} y^{2} y^{\prime }-4 x^{2} y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.287 (sec). Leaf size: 132

ode:=diff(y(x),x)^2-2*x^3*y(x)^2*diff(y(x),x)-4*x^2*y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {4}{x^{4}} \\ y &= 0 \\ y &= \frac {\left (\sqrt {2}\, x^{2} c_1 -2\right ) c_1^{2}}{2 c_1^{2} x^{4}-4} \\ y &= -\frac {\left (\sqrt {2}\, x^{2} c_1 +2\right ) c_1^{2}}{2 c_1^{2} x^{4}-4} \\ y &= \frac {-2 \sqrt {2}\, x^{2}+2 c_1}{c_1 \left (-2 x^{4}+c_1^{2}\right )} \\ y &= \frac {2 \sqrt {2}\, x^{2}+2 c_1}{c_1 \left (-2 x^{4}+c_1^{2}\right )} \\ \end{align*}

✓ Mathematica. Time used: 2.828 (sec). Leaf size: 179

ode=-4*x^2*y[x]^3 - 2*x^3*y[x]^2*D[y[x],x] + D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [-\frac {x \sqrt {x^4 y(x)+4} y(x)^{3/2} \log \left (\sqrt {x^4 y(x)+4}-x^2 \sqrt {y(x)}\right )}{2 \sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}-\frac {1}{4} \log (y(x))&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {x y(x)^{3/2} \sqrt {x^4 y(x)+4} \log \left (\sqrt {x^4 y(x)+4}-x^2 \sqrt {y(x)}\right )}{2 \sqrt {x^2 y(x)^3 \left (x^4 y(x)+4\right )}}-\frac {1}{4} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to 0 \\ y(x)\to -\frac {4}{x^4} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x*(x**2*y(x)**2 + sqrt((x**4*y(x) + 4)*y(x)**3)) + Derivative(y(x), x) cannot be solved by the factorable group method