Internal problem ID [3459]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 721.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y^{\prime } \sqrt {b^{2}+y^{2}}-\sqrt {a^{2}+x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 67
dsolve(diff(y(x),x)*sqrt(b^2+y(x)^2) = sqrt(a^2+x^2),y(x), singsol=all)
\[ \frac {x \sqrt {a^{2}+x^{2}}}{2}+\frac {a^{2} \ln \left (x +\sqrt {a^{2}+x^{2}}\right )}{2}-\frac {y \relax (x ) \sqrt {b^{2}+y \relax (x )^{2}}}{2}-\frac {b^{2} \ln \left (y \relax (x )+\sqrt {b^{2}+y \relax (x )^{2}}\right )}{2}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.672 (sec). Leaf size: 89
DSolve[y'[x] Sqrt[y[x]^2+b^2]==Sqrt[x^2+a^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{2} \text {$\#$1} \sqrt {\text {$\#$1}^2+b^2}+\frac {1}{2} b^2 \tanh ^{-1}\left (\frac {\text {$\#$1}}{\sqrt {\text {$\#$1}^2+b^2}}\right )\&\right ]\left [\frac {1}{2} \left (x \sqrt {a^2+x^2}+a^2 \tanh ^{-1}\left (\frac {x}{\sqrt {a^2+x^2}}\right )+2 c_1\right )\right ] \\ \end{align*}