29.14.13 problem 394

Internal problem ID [4992]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 14
Problem number : 394
Date solved : Monday, January 27, 2025 at 10:01:44 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 37

dsolve(diff(y(x),x)*sqrt(b^2-x^2) = sqrt(a^2-y(x)^2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 4.742 (sec). Leaf size: 118

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

\begin{align*} y(x)\to \frac {a \tan \left (\arctan \left (\frac {x}{\sqrt {b^2-x^2}}\right )+c_1\right )}{\sqrt {\sec ^2\left (\arctan \left (\frac {x}{\sqrt {b^2-x^2}}\right )+c_1\right )}} \\ y(x)\to -\frac {a \tan \left (\arctan \left (\frac {x}{\sqrt {b^2-x^2}}\right )+c_1\right )}{\sqrt {\sec ^2\left (\arctan \left (\frac {x}{\sqrt {b^2-x^2}}\right )+c_1\right )}} \\ y(x)\to -a \\ y(x)\to a \\ \end{align*}