60.1.408 problem 419

Internal problem ID [10422]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 419
Date solved : Sunday, March 30, 2025 at 04:41:27 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 111

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

\begin{align*} x -\frac {2^{{1}/{3}} \left (-y+\sqrt {x^{2}+y^{2}}\right ) c_1}{2 \left (\frac {3 y^{2}-3 y \sqrt {x^{2}+y^{2}}+x^{2}}{x^{2}}\right )^{{2}/{3}} x} &= 0 \\ \frac {c_1 \left (y+\sqrt {x^{2}+y^{2}}\right )}{{\left (\frac {3 y \sqrt {x^{2}+y^{2}}+x^{2}+3 y^{2}}{x^{2}}\right )}^{{2}/{3}} x}+x &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.651 (sec). Leaf size: 6977

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

Too large to display

✓ Sympy. Time used: 11.246 (sec). Leaf size: 75

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

\[ \left [ \log {\left (x \right )} = C_{1} - \log {\left (\left (\sqrt {1 + \frac {y^{2}{\left (x \right )}}{x^{2}}} + \frac {2 y{\left (x \right )}}{x}\right )^{\frac {2}{3}} \right )} + \frac {\operatorname {asinh}{\left (\frac {y{\left (x \right )}}{x} \right )}}{3}, \ \log {\left (x \right )} = C_{1} - \log {\left (\left (- \sqrt {1 + \frac {y^{2}{\left (x \right )}}{x^{2}}} + \frac {2 y{\left (x \right )}}{x}\right )^{\frac {2}{3}} \right )} - \frac {\operatorname {asinh}{\left (\frac {y{\left (x \right )}}{x} \right )}}{3}\right ] \]