Internal problem ID [3650]
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]
\[ \boxed {x y^{\prime } \sqrt {-a^{2}+x^{2}}-y \sqrt {y^{2}-b^{2}}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 135
dsolve(x*diff(y(x),x)*sqrt(-a^2+x^2) = y(x)*sqrt(y(x)^2-b^2),y(x), singsol=all)
\[ \frac {c_{1} \sqrt {-a^{2}}\, \sqrt {-b^{2}}+\sqrt {-a^{2}}\, \ln \left (2\right )-\sqrt {-b^{2}}\, \ln \left (2\right )-\sqrt {-b^{2}}\, \ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )+\sqrt {-a^{2}}\, \ln \left (\frac {\sqrt {-b^{2}}\, \sqrt {y \left (x \right )^{2}-b^{2}}-b^{2}}{y \left (x \right )}\right )}{\sqrt {-a^{2}}\, \sqrt {-b^{2}}} = 0 \]
✓ Solution by Mathematica
Time used: 18.348 (sec). Leaf size: 101
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 (\frac {b \left (\arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )+a c_1\right )}{a}\right )} \\ y(x)\to b \sqrt {\sec ^2\left (\frac {b \left (\arctan \left (\frac {\sqrt {x^2-a^2}}{a}\right )+a c_1\right )}{a}\right )} \\ y(x)\to 0 \\ y(x)\to -b \\ y(x)\to b \\ \end{align*}