69.1.35 problem 53
Internal
problem
ID
[14117]
Book
:
DIFFERENTIAL
and
INTEGRAL
CALCULUS.
VOL
I.
by
N.
PISKUNOV.
MIR
PUBLISHERS,
Moscow
1969.
Section
:
Chapter
8.
Differential
equations.
Exercises
page
595
Problem
number
:
53
Date
solved
:
Monday, March 31, 2025 at 12:06:56 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _dAlembert]
\begin{align*} \frac {x +y y^{\prime }}{\sqrt {x^{2}+y^{2}}}&=m \end{align*}
✓ Maple. Time used: 0.099 (sec). Leaf size: 183
ode:=(x+y(x)*diff(y(x),x))/(x^2+y(x)^2)^(1/2) = m;
dsolve(ode,y(x), singsol=all);
\[
-\int _{\textit {\_b}}^{x}\frac {m \sqrt {\textit {\_a}^{2}+y^{2}}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+y^{2}}\, m \textit {\_a} -\textit {\_a}^{2}-y^{2}}d \textit {\_a} -\int _{}^{y}\frac {\textit {\_f} \left (-1+\left (\sqrt {\textit {\_f}^{2}+x^{2}}\, m x -x^{2}-\textit {\_f}^{2}\right ) \int _{\textit {\_b}}^{x}-\frac {2 \left (-\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \textit {\_a} +m \left (\textit {\_a}^{2}+\frac {\textit {\_f}^{2}}{2}\right )\right )}{\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \left (\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, m \textit {\_a} -\textit {\_a}^{2}-\textit {\_f}^{2}\right )^{2}}d \textit {\_a} \right )}{\sqrt {\textit {\_f}^{2}+x^{2}}\, m x -x^{2}-\textit {\_f}^{2}}d \textit {\_f} +c_1 = 0
\]
✓ Mathematica. Time used: 2.146 (sec). Leaf size: 103
ode=(x+y[x]*D[y[x],x])/Sqrt[x^2+y[x]^2]==m;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\sqrt {\left (m^2-1\right ) x^2-2 e^{c_1} m x+e^{2 c_1}} \\
y(x)\to \sqrt {\left (m^2-1\right ) x^2-2 e^{c_1} m x+e^{2 c_1}} \\
y(x)\to -\sqrt {\left (m^2-1\right ) x^2} \\
y(x)\to \sqrt {\left (m^2-1\right ) x^2} \\
\end{align*}
✓ Sympy. Time used: 92.382 (sec). Leaf size: 354
from sympy import *
x = symbols("x")
m = symbols("m")
y = Function("y")
ode = Eq(-m + (x + y(x)*Derivative(y(x), x))/sqrt(x**2 + y(x)**2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 m^{2} x^{2} - 2 \sqrt {2} m x \sqrt {- C_{1}} - 2 x^{2}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 m^{2} x^{2} - 2 \sqrt {2} m x \sqrt {- C_{1}} - 2 x^{2}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 m^{2} x^{2} + 2 \sqrt {2} m x \sqrt {- C_{1}} - 2 x^{2}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 m^{2} x^{2} + 2 \sqrt {2} m x \sqrt {- C_{1}} - 2 x^{2}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} m x + C_{1} + 2 m^{2} x^{2} - 2 x^{2}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} m x + C_{1} + 2 m^{2} x^{2} - 2 x^{2}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} m x + C_{1} + 2 m^{2} x^{2} - 2 x^{2}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} m x + C_{1} + 2 m^{2} x^{2} - 2 x^{2}}}{2}\right ]
\]