29.31.21 problem 920

Internal problem ID [5500]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 31
Problem number : 920
Date solved : Monday, January 27, 2025 at 11:32:27 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

\begin{align*} \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}+2 x y y^{\prime }+x^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 1.415 (sec). Leaf size: 51

dsolve((a^2-x^2)*diff(y(x),x)^2+2*x*y(x)*diff(y(x),x)+x^2 = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \sqrt {a^{2}-x^{2}} \\ y \left (x \right ) &= -\sqrt {a^{2}-x^{2}} \\ y \left (x \right ) &= c_{1} x^{2}-c_{1} a^{2}-\frac {1}{4 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.413 (sec). Leaf size: 67

DSolve[(a^2-x^2) (D[y[x],x])^2+2 x y[x] D[y[x],x]+x^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {a^2-x^2+c_1{}^2}{2 c_1} \\ y(x)\to \text {Indeterminate} \\ y(x)\to -\sqrt {a^2-x^2} \\ y(x)\to \sqrt {a^2-x^2} \\ \end{align*}