29.14.14 problem 395
Internal
problem
ID
[4993]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
14
Problem
number
:
395
Date
solved
:
Sunday, March 30, 2025 at 04:29:21 AM
CAS
classification
:
[_separable]
\begin{align*} x y^{\prime } \sqrt {a^{2}+x^{2}}&=y \sqrt {b^{2}+y^{2}} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 79
ode:=x*diff(y(x),x)*(a^2+x^2)^(1/2) = y(x)*(b^2+y(x)^2)^(1/2);
dsolve(ode,y(x), singsol=all);
\[
\frac {-\operatorname {csgn}\left (a \right ) b \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}+x^{2}}+a \right )}{x}\right )+\operatorname {csgn}\left (b \right ) a \ln \left (\frac {b \left (\operatorname {csgn}\left (b \right ) \sqrt {b^{2}+y^{2}}+b \right )}{y}\right )+\operatorname {csgn}\left (b \right ) a \ln \left (2\right )-\operatorname {csgn}\left (a \right ) b \ln \left (2\right )+c_1 a b}{a b} = 0
\]
✓ Mathematica. Time used: 44.311 (sec). Leaf size: 276
ode=x D[y[x],x] Sqrt[a^2+x^2]==y[x] Sqrt[b^2+y[x]^2];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {2 i b^{3/2} e^{b c_1} \left (a \left (a-\sqrt {a^2+x^2}\right )\right )^{\frac {b}{2 a}} \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{2 a}}}{\sqrt {\left (b \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{a}}+e^{2 b c_1} \left (a \left (a-\sqrt {a^2+x^2}\right )\right )^{\frac {b}{a}}\right ){}^2}} \\
y(x)\to \frac {2 i b^{3/2} e^{b c_1} \left (a \left (a-\sqrt {a^2+x^2}\right )\right )^{\frac {b}{2 a}} \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{2 a}}}{\sqrt {\left (b \left (\sqrt {a^2+x^2}+a\right )^{\frac {b}{a}}+e^{2 b c_1} \left (a \left (a-\sqrt {a^2+x^2}\right )\right )^{\frac {b}{a}}\right ){}^2}} \\
y(x)\to 0 \\
y(x)\to -i b \\
y(x)\to i b \\
\end{align*}
✓ Sympy. Time used: 1.255 (sec). Leaf size: 17
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(x*sqrt(a**2 + x**2)*Derivative(y(x), x) - sqrt(b**2 + y(x)**2)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = - \frac {b}{\sinh {\left (b \left (C_{1} - \frac {\operatorname {asinh}{\left (\frac {a}{x} \right )}}{a}\right ) \right )}}
\]