29.29.22 problem 844

Internal problem ID [5427]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 29
Problem number : 844
Date solved : Monday, January 27, 2025 at 11:23:18 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.040 (sec). Leaf size: 47

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

\begin{align*} y \left (x \right ) &= \int \frac {\sqrt {x \left (-x^{2}+a \right )}}{x}d x +c_{1} \\ y \left (x \right ) &= -\int \frac {\sqrt {x \left (-x^{2}+a \right )}}{x}d x +c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 113

DSolve[x (D[y[x],x])^2==(a-x^2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {2 \sqrt {x} \sqrt {a-x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {x^2}{a}\right )}{\sqrt {1-\frac {x^2}{a}}}+c_1 \\ y(x)\to \frac {2 \sqrt {x} \sqrt {a-x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {x^2}{a}\right )}{\sqrt {1-\frac {x^2}{a}}}+c_1 \\ \end{align*}