60.1.545 problem 558
Internal
problem
ID
[10559]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
558
Date
solved
:
Wednesday, March 05, 2025 at 12:05:04 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} a x \sqrt {{y^{\prime }}^{2}+1}+x y^{\prime }-y&=0 \end{align*}
✓ Maple. Time used: 0.098 (sec). Leaf size: 340
ode:=a*x*(1+diff(y(x),x)^2)^(1/2)+x*diff(y(x),x)-y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a +y}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\
\frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y^{2}-2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {-\sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a +y}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y^{2}-2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 1.687 (sec). Leaf size: 223
ode=-y[x] + x*D[y[x],x] + a*x*Sqrt[1 + D[y[x],x]^2]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\
\text {Solve}\left [\frac {-2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\
\end{align*}
✓ Sympy. Time used: 29.657 (sec). Leaf size: 144
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*x*sqrt(Derivative(y(x), x)**2 + 1) + x*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {- u_{1}^{2} a^{2} + u_{1}^{2} + 1}}{u_{1} \left (a + \sqrt {- u_{1}^{2} a^{2} + u_{1}^{2} + 1}\right )}\, du_{1} - \frac {\int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (a + \sqrt {- u_{1}^{2} a^{2} + u_{1}^{2} + 1}\right )}\, du_{1}}{a}}, \ y{\left (x \right )} = C_{1} e^{\int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {- u_{1}^{2} a^{2} + u_{1}^{2} + 1}}{u_{1} \left (a - \sqrt {- u_{1}^{2} a^{2} + u_{1}^{2} + 1}\right )}\, du_{1} - \frac {\int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (a - \sqrt {- u_{1}^{2} a^{2} + u_{1}^{2} + 1}\right )}\, du_{1}}{a}}\right ]
\]