29.26.1 problem 732
Internal
problem
ID
[5322]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
26
Problem
number
:
732
Date
solved
:
Tuesday, March 04, 2025 at 09:25:43 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} x y \left (x +\sqrt {x^{2}-y^{2}}\right ) y^{\prime }&=x y^{2}-\left (x^{2}-y^{2}\right )^{{3}/{2}} \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 40
ode:=x*y(x)*(x+(x^2-y(x)^2)^(1/2))*diff(y(x),x) = x*y(x)^2-(x^2-y(x)^2)^(3/2);
dsolve(ode,y(x), singsol=all);
\[
\frac {2 \ln \left (x \right ) x^{2}-c_{1} x^{2}+y \left (x \right )^{2}-2 x \sqrt {x^{2}-y \left (x \right )^{2}}}{x^{2}} = 0
\]
✓ Mathematica. Time used: 22.831 (sec). Leaf size: 385
ode=x*y[x]*(x+Sqrt[x^2-y[x]^2])*D[y[x],x]==x*y[x]^2-(x^2-y[x]^2)^(3/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\sqrt {-2 \sqrt {-x^4 (-2 \log (x)-1+2 c_1)}-2 x^2 \log (x)+(-1+2 c_1) x^2} \\
y(x)\to \sqrt {-2 \sqrt {-x^4 (-2 \log (x)-1+2 c_1)}-2 x^2 \log (x)+(-1+2 c_1) x^2} \\
y(x)\to -\sqrt {2 \sqrt {-x^4 (-2 \log (x)-1+2 c_1)}-2 x^2 \log (x)+(-1+2 c_1) x^2} \\
y(x)\to \sqrt {2 \sqrt {-x^4 (-2 \log (x)-1+2 c_1)}-2 x^2 \log (x)+(-1+2 c_1) x^2} \\
y(x)\to -\sqrt {-2 \sqrt {x^4 (-2 \log (x)+1+2 c_1)}+2 x^2 \log (x)-\left ((1+2 c_1) x^2\right )} \\
y(x)\to \sqrt {-2 \sqrt {x^4 (-2 \log (x)+1+2 c_1)}+2 x^2 \log (x)-\left ((1+2 c_1) x^2\right )} \\
y(x)\to -\sqrt {2 \sqrt {x^4 (-2 \log (x)+1+2 c_1)}+2 x^2 \log (x)-\left ((1+2 c_1) x^2\right )} \\
y(x)\to \sqrt {2 \sqrt {x^4 (-2 \log (x)+1+2 c_1)}+2 x^2 \log (x)-\left ((1+2 c_1) x^2\right )} \\
\end{align*}
✓ Sympy. Time used: 1.370 (sec). Leaf size: 70
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(x + sqrt(x**2 - y(x)**2))*y(x)*Derivative(y(x), x) - x*y(x)**2 + (x**2 - y(x)**2)**(3/2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\log {\left (y{\left (x \right )} \right )} = C_{1} + \begin {cases} \frac {\sqrt {\frac {x^{2}}{y^{2}{\left (x \right )}} - 1} y{\left (x \right )}}{x} & \text {for}\: \left |{\frac {x^{2}}{y^{2}{\left (x \right )}}}\right | > 1 \\\frac {i \sqrt {- \frac {x^{2}}{y^{2}{\left (x \right )}} + 1} y{\left (x \right )}}{x} & \text {otherwise} \end {cases} - \log {\left (\frac {x}{y{\left (x \right )}} \right )} - \frac {y^{2}{\left (x \right )}}{2 x^{2}}
\]