60.1.404 problem 415

Internal problem ID [10418]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 415
Date solved : Wednesday, March 05, 2025 at 10:47:24 AM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 89

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

\begin{align*} y &= -\frac {1}{2 \sqrt {-x}} \\ y &= \frac {1}{2 \sqrt {-x}} \\ y &= 0 \\ y &= -\frac {\coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right ) \sqrt {\operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )^{2} x}}{2 x} \\ y &= \frac {\coth \left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right ) \sqrt {\operatorname {sech}\left (-\frac {\ln \left (x \right )}{2}+\frac {c_{1}}{2}\right )^{2} x}}{2 x} \\ \end{align*}

✓ Mathematica. Time used: 0.525 (sec). Leaf size: 84

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

\begin{align*} y(x)\to -\frac {2 e^{\frac {c_1}{2}}}{-4 x+e^{c_1}} \\ y(x)\to \frac {2 e^{\frac {c_1}{2}}}{-4 x+e^{c_1}} \\ y(x)\to 0 \\ y(x)\to -\frac {i}{2 \sqrt {x}} \\ y(x)\to \frac {i}{2 \sqrt {x}} \\ \end{align*}

✓ Sympy. Time used: 18.360 (sec). Leaf size: 39

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

\[ \left [ y{\left (x \right )} = \frac {1}{2 \sqrt {x} \sinh {\left (C_{1} + \frac {\log {\left (x \right )}}{2} \right )}}, \ y{\left (x \right )} = - \frac {1}{2 \sqrt {x} \sinh {\left (C_{1} + \frac {\log {\left (x \right )}}{2} \right )}}\right ] \]