60.1.457 problem 470

Internal problem ID [10471]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 470
Date solved : Sunday, March 30, 2025 at 04:51:02 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.316 (sec). Leaf size: 87

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

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

✓ Mathematica. Time used: 1.108 (sec). Leaf size: 178

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

\begin{align*} \text {Solve}\left [\frac {1}{2} \log (y(x))-\frac {\sqrt {x^6+4 x^2 y(x)^2} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+4 y(x)^2}+2 y(x)}\right )}{x \sqrt {x^4+4 y(x)^2}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {\sqrt {x^6+4 x^2 y(x)^2} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+4 y(x)^2}+2 y(x)}\right )}{x \sqrt {x^4+4 y(x)^2}}+\frac {1}{2} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to -\frac {i x^2}{2} \\ y(x)\to \frac {i x^2}{2} \\ \end{align*}

✗ Sympy

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

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