83.23.17 problem 17

Internal problem ID [19298]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 01:24:46 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Solution by Maple

Time used: 0.493 (sec). Leaf size: 51

dsolve((x^2-a^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: 0.404 (sec). Leaf size: 67

DSolve[(x^2-a^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*}