Internal problem ID [3460]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 25
Problem number: 722.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y^{\prime } \sqrt {-y^{2}+b^{2}}-\sqrt {a^{2}-x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 75
dsolve(diff(y(x),x)*sqrt(b^2-y(x)^2) = sqrt(a^2-x^2),y(x), singsol=all)
\[ \frac {x \sqrt {a^{2}-x^{2}}}{2}+\frac {a^{2} \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )}{2}-\frac {y \relax (x ) \sqrt {b^{2}-y \relax (x )^{2}}}{2}-\frac {b^{2} \arctan \left (\frac {y \relax (x )}{\sqrt {b^{2}-y \relax (x )^{2}}}\right )}{2}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.746 (sec). Leaf size: 97
DSolve[y'[x] Sqrt[b^2-y[x]^2]==Sqrt[a^2-x^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{2} b^2 \text {ArcTan}\left (\frac {\text {$\#$1}}{\sqrt {b^2-\text {$\#$1}^2}}\right )+\frac {1}{2} \text {$\#$1} \sqrt {b^2-\text {$\#$1}^2}\&\right ]\left [\frac {1}{2} \left (a^2 \text {ArcTan}\left (\frac {x}{\sqrt {a^2-x^2}}\right )+x \sqrt {a^2-x^2}+2 c_1\right )\right ] \\ \end{align*}