83.22.7 problem 7
Internal
problem
ID
[19190]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
IV.
Equations
of
the
first
order
but
not
of
the
first
degree.
Exercise
IV
(E)
at
page
63
Problem
number
:
7
Date
solved
:
Monday, March 31, 2025 at 06:52:26 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries]]
\begin{align*} \left (x y^{\prime }-y\right )^{2}&=a \left (1+{y^{\prime }}^{2}\right ) \left (x^{2}+y^{2}\right )^{{3}/{2}} \end{align*}
✓ Maple. Time used: 7.300 (sec). Leaf size: 98
ode:=(-y(x)+x*diff(y(x),x))^2 = a*(1+diff(y(x),x)^2)*(x^2+y(x)^2)^(3/2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -i x \\
y &= i x \\
y &= x \cot \left (\operatorname {RootOf}\left (-4 \textit {\_Z} -2 \int _{}^{x^{2} \csc \left (\textit {\_Z} \right )^{2}}\frac {\sqrt {-\textit {\_a}^{{17}/{2}} \left (\sqrt {\textit {\_a}}\, a -1\right ) \left (2 \sqrt {\textit {\_a}}\, a +\cos \left (2\right )-1\right )^{2} a}}{\textit {\_a}^{5} \left (2 \textit {\_a} \,a^{2}-3 \sqrt {\textit {\_a}}\, a +1+\sqrt {\textit {\_a}}\, a \cos \left (2\right )-\cos \left (2\right )\right )}d \textit {\_a} +4 c_1 \right )\right ) \\
\end{align*}
✓ Mathematica. Time used: 46.576 (sec). Leaf size: 305
ode=(x*D[y[x],x]-y[x])^2==a*(1+D[y[x],x]^2)*(x^2+y[x]^2)^(3/2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\arctan \left (\frac {x}{y(x)}\right )-\frac {2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}\right )} \arctan \left (\frac {\sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}{\sqrt {a} \sqrt {x^2+y(x)^2}}\right )}{\sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {2 \sqrt {a \left (x^2+y(x)^2\right )^2 \left (-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}\right )} \arctan \left (\frac {\sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}{\sqrt {a} \sqrt {x^2+y(x)^2}}\right )}{\sqrt {a} \left (x^2+y(x)^2\right ) \sqrt {-a x^2-a y(x)^2+\sqrt {x^2+y(x)^2}}}+\arctan \left (\frac {x}{y(x)}\right )&=c_1,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*(x**2 + y(x)**2)**(3/2)*(Derivative(y(x), x)**2 + 1) + (x*Derivative(y(x), x) - y(x))**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out