29.25.25 problem 722

Internal problem ID [5312]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 25
Problem number : 722
Date solved : Monday, January 27, 2025 at 11:08:27 AM
CAS classification : [_separable]

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

✓ 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)
 

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

✓ Solution by Mathematica

Time used: 0.894 (sec). Leaf size: 94

DSolve[D[y[x],x] Sqrt[b^2-y[x]^2]==Sqrt[a^2-x^2],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \text {InverseFunction}\left [\frac {1}{2} \left (b^2 \arctan \left (\frac {\text {$\#$1}}{\sqrt {b^2-\text {$\#$1}^2}}\right )+\text {$\#$1} \sqrt {b^2-\text {$\#$1}^2}\right )\&\right ]\left [\frac {1}{2} \left (a^2 \arctan \left (\frac {x}{\sqrt {a^2-x^2}}\right )+x \sqrt {a^2-x^2}\right )+c_1\right ] \]