77.1.58 problem 77 (page 120)
Internal
problem
ID
[17877]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
77
(page
120)
Date
solved
:
Monday, March 31, 2025 at 04:43:52 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y&=\frac {k \left (x +y y^{\prime }\right )}{\sqrt {1+{y^{\prime }}^{2}}} \end{align*}
✓ Maple. Time used: 0.672 (sec). Leaf size: 378
ode:=y(x) = k*(x+y(x)*diff(y(x),x))/(1+diff(y(x),x)^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
-{\mathrm e}^{k \int _{}^{\frac {-k^{2} x +\sqrt {y^{2} \left (k^{2}-1\right )+k^{2} x^{2}}}{\left (k^{2}-1\right ) y}}\frac {k \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (k \textit {\_a} -\sqrt {\textit {\_a}^{2}+1}\right ) \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +k \left (\textit {\_a}^{2}+1\right )\right )}d \textit {\_a}} c_1 +x &= 0 \\
-{\mathrm e}^{k \int _{}^{\frac {-k^{2} x -\sqrt {y^{2} \left (k^{2}-1\right )+k^{2} x^{2}}}{y \left (k^{2}-1\right )}}\frac {k \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (k \textit {\_a} -\sqrt {\textit {\_a}^{2}+1}\right ) \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +k \left (\textit {\_a}^{2}+1\right )\right )}d \textit {\_a}} c_1 +x &= 0 \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\left (\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}-\sqrt {\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}\right )}d \textit {\_a} +c_1 \right ) x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}+\sqrt {\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} k^{2}-\textit {\_a}^{2}+k^{2}\right )}d \textit {\_a} +c_1 \right ) x \\
\end{align*}
✗ Mathematica
ode=y[x]==k*(x+y[x]*D[y[x],x] )/sqrt(1+D[y[x],x]^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Timed out
✗ Sympy
from sympy import *
x = symbols("x")
k = symbols("k")
y = Function("y")
ode = Eq(-k*(x + y(x)*Derivative(y(x), x))/sqrt(Derivative(y(x), x)**2 + 1) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out