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
:
Monday, January 27, 2025 at 10:01:58 AM
CAS
classification
:
[_separable]
\begin{align*} x y^{\prime } \sqrt {a^{2}+x^{2}}&=y \sqrt {b^{2}+y^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 79
dsolve(x*diff(y(x),x)*sqrt(a^2+x^2) = y(x)*sqrt(b^2+y(x)^2),y(x), singsol=all)
\[
\frac {-\operatorname {csgn}\left (a \right ) b \ln \left (2\right )-\operatorname {csgn}\left (a \right ) b \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}+x^{2}}+a \right )}{x}\right )+\operatorname {csgn}\left (b \right ) a \ln \left (2\right )+\operatorname {csgn}\left (b \right ) a \ln \left (\frac {b \left (\operatorname {csgn}\left (b \right ) \sqrt {b^{2}+y \left (x \right )^{2}}+b \right )}{y \left (x \right )}\right )+c_{1} a b}{a b} = 0
\]
✓ Solution by Mathematica
Time used: 44.311 (sec). Leaf size: 276
DSolve[x D[y[x],x] Sqrt[a^2+x^2]==y[x] Sqrt[b^2+y[x]^2],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*}