Internal problem ID [3142]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 14
Problem number: 396.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {x y^{\prime } \sqrt {-a^{2}+x^{2}}-y \sqrt {y^{2}-b^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 86
dsolve(x*diff(y(x),x)*sqrt(-a^2+x^2) = y(x)*sqrt(y(x)^2-b^2),y(x), singsol=all)
\[ -\frac {\ln \left (\frac {-2 a^{2}+2 \sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}}{x}\right )}{\sqrt {-a^{2}}}+\frac {\ln \left (\frac {-2 b^{2}+2 \sqrt {-b^{2}}\, \sqrt {y \relax (x )^{2}-b^{2}}}{y \relax (x )}\right )}{\sqrt {-b^{2}}}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 17.839 (sec). Leaf size: 95
DSolve[x y'[x] Sqrt[x^2-a^2]==y[x] Sqrt[y[x]^2-b^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -b \sqrt {\sec ^2\left (b \left (\frac {\cot ^{-1}\left (\frac {a}{\sqrt {x^2-a^2}}\right )}{a}+c_1\right )\right )} \\ y(x)\to b \sqrt {\sec ^2\left (b \left (\frac {\cot ^{-1}\left (\frac {a}{\sqrt {x^2-a^2}}\right )}{a}+c_1\right )\right )} \\ y(x)\to 0 \\ y(x)\to -b \\ y(x)\to b \\ \end{align*}