29.14.15 problem 396

Internal problem ID [4994]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 396
Date solved : Monday, January 27, 2025 at 10:02:00 AM
CAS classification : [_separable]

\begin{align*} x y^{\prime } \sqrt {-a^{2}+x^{2}}&=y \sqrt {y^{2}-b^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.003 (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)
 

\[ \frac {c_{1} \sqrt {-a^{2}}\, \sqrt {-b^{2}}-\sqrt {-b^{2}}\, \ln \left (2\right )+\sqrt {-a^{2}}\, \ln \left (2\right )-\sqrt {-b^{2}}\, \ln \left (\frac {\sqrt {-a^{2}}\, \sqrt {-a^{2}+x^{2}}-a^{2}}{x}\right )+\sqrt {-a^{2}}\, \ln \left (\frac {\sqrt {-b^{2}}\, \sqrt {y \left (x \right )^{2}-b^{2}}-b^{2}}{y \left (x \right )}\right )}{\sqrt {-a^{2}}\, \sqrt {-b^{2}}} = 0 \]

✓ Solution by Mathematica

Time used: 18.123 (sec). Leaf size: 101

DSolve[x D[y[x],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*}