60.1.527 problem 540
Internal
problem
ID
[10541]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
540
Date
solved
:
Sunday, March 30, 2025 at 05:49:09 PM
CAS
classification
:
[_quadrature]
\begin{align*} 2 y {y^{\prime }}^{3}-y {y^{\prime }}^{2}+2 x y^{\prime }-x&=0 \end{align*}
✓ Maple. Time used: 0.033 (sec). Leaf size: 111
ode:=2*y(x)*diff(y(x),x)^3-y(x)*diff(y(x),x)^2+2*x*diff(y(x),x)-x = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
x \left (1+\frac {c_1}{\left (\frac {-\sqrt {-y x}+y}{y}\right )^{{2}/{3}} \left (\frac {-x +\sqrt {-y x}+y}{y}\right )^{{2}/{3}} y}\right ) &= 0 \\
x \left (1+\frac {c_1}{\left (\frac {\sqrt {-y x}+y}{y}\right )^{{2}/{3}} \left (\frac {-x -\sqrt {-y x}+y}{y}\right )^{{2}/{3}} y}\right ) &= 0 \\
y &= \frac {x}{2}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 3.466 (sec). Leaf size: 61
ode=-x + 2*x*D[y[x],x] - y[x]*D[y[x],x]^2 + 2*y[x]*D[y[x],x]^3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {x}{2}+c_1 \\
y(x)\to \left (\frac {3 c_1}{2}-i x^{3/2}\right ){}^{2/3} \\
y(x)\to \left (i x^{3/2}+\frac {3 c_1}{2}\right ){}^{2/3} \\
\end{align*}
✓ Sympy. Time used: 74.506 (sec). Leaf size: 381
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*Derivative(y(x), x) - x + 2*y(x)*Derivative(y(x), x)**3 - y(x)*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} + \frac {x}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}\right ]
\]