29.33.7 problem 969
Internal
problem
ID
[5546]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
33
Problem
number
:
969
Date
solved
:
Sunday, March 30, 2025 at 08:33:10 AM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} x \left (x -2 y\right ) {y^{\prime }}^{2}+6 x y y^{\prime }-2 x y+y^{2}&=0 \end{align*}
✓ Maple. Time used: 0.069 (sec). Leaf size: 115
ode:=x*(x-2*y(x))*diff(y(x),x)^2+6*x*y(x)*diff(y(x),x)-2*x*y(x)+y(x)^2 = 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: 5.807 (sec). Leaf size: 196
ode=x(x-2 y[x]) (D[y[x],x])^2+6 x y[x] D[y[x],x]-2 x y[x]+y[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: 13.576 (sec). Leaf size: 75
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x - 2*y(x))*Derivative(y(x), x)**2 + 6*x*y(x)*Derivative(y(x), x) - 2*x*y(x) + y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} + 2 x - \sqrt {x \left (2 C_{1} + 3 x\right )}, \ y{\left (x \right )} = C_{1} + 2 x + \sqrt {x \left (2 C_{1} + 3 x\right )}, \ y{\left (x \right )} = C_{1} + 2 x - \sqrt {x \left (2 C_{1} + 3 x\right )}, \ y{\left (x \right )} = C_{1} + 2 x + \sqrt {x \left (2 C_{1} + 3 x\right )}\right ]
\]