60.1.447 problem 450

Internal problem ID [10461]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 450
Date solved : Monday, January 27, 2025 at 07:46:11 PM
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: 1.102 (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 &= \sqrt {a^{2}-x^{2}} \\ y &= -\sqrt {a^{2}-x^{2}} \\ y &= c_{1} x^{2}-c_{1} a^{2}-\frac {1}{4 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.417 (sec). Leaf size: 67

DSolve[-x^2 - 2*x*y[x]*D[y[x],x] + (-a^2 + x^2)*D[y[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*}