60.1.471 problem 484
Internal
problem
ID
[10485]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
484
Date
solved
:
Sunday, March 30, 2025 at 05:01:16 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} \left (2 x y-x^{2}\right ) {y^{\prime }}^{2}-6 x y y^{\prime }-y^{2}+2 x y&=0 \end{align*}
✓ Maple. Time used: 0.058 (sec). Leaf size: 115
ode:=(2*x*y(x)-x^2)*diff(y(x),x)^2-6*x*y(x)*diff(y(x),x)-y(x)^2+2*x*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} +1\right )^{2}}-4 \textit {\_a}}{\textit {\_a} \left (\textit {\_a}^{2}-4 \textit {\_a} +1\right )}d \textit {\_a} +2 c_1 \right ) x \\
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} +1\right )^{2}}-2 \textit {\_a}^{2}+4 \textit {\_a}}{\textit {\_a} \left (\textit {\_a}^{2}-4 \textit {\_a} +1\right )}d \textit {\_a} +2 c_1 \right ) x \\
\end{align*}
✓ Mathematica. Time used: 5.8 (sec). Leaf size: 196
ode=2*x*y[x] - y[x]^2 - 6*x*y[x]*D[y[x],x] + (-x^2 + 2*x*y[x])*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to 2 x-\sqrt {x \left (3 x-2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}} \\
y(x)\to 2 x+\sqrt {x \left (3 x-2 e^{\frac {c_1}{2}}\right )}-e^{\frac {c_1}{2}} \\
y(x)\to 2 x-\sqrt {x \left (3 x+2 e^{\frac {c_1}{2}}\right )}+e^{\frac {c_1}{2}} \\
y(x)\to 2 x+\sqrt {x \left (3 x+2 e^{\frac {c_1}{2}}\right )}+e^{\frac {c_1}{2}} \\
y(x)\to 2 x-\sqrt {3} \sqrt {x^2} \\
y(x)\to \sqrt {3} \sqrt {x^2}+2 x \\
\end{align*}
✓ Sympy. Time used: 13.237 (sec). Leaf size: 75
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-6*x*y(x)*Derivative(y(x), x) + 2*x*y(x) + (-x**2 + 2*x*y(x))*Derivative(y(x), x)**2 - y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} + 2 x - \sqrt {x \left (2 C_{1} + 3 x\right )}, \ y{\left (x \right )} = C_{1} + 2 x + \sqrt {x \left (2 C_{1} + 3 x\right )}, \ y{\left (x \right )} = C_{1} + 2 x - \sqrt {x \left (2 C_{1} + 3 x\right )}, \ y{\left (x \right )} = C_{1} + 2 x + \sqrt {x \left (2 C_{1} + 3 x\right )}\right ]
\]