83.22.15 problem 15

Internal problem ID [19269]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 01:18:12 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} {y^{\prime }}^{2} \left (-a^{2}+x^{2}\right )-2 x y y^{\prime }+y^{2}+a^{4}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.083 (sec). Leaf size: 80

dsolve(diff(y(x),x)^2*(x^2-a^2)-2*diff(y(x),x)*x*y(x)+y(x)^2+a^4=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.479 (sec). Leaf size: 101

DSolve[D[y[x],x]^2*(x^2-a^2)-2*D[y[x],x]*x*y[x]+y[x]^2+a^4==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_1 x-\sqrt {-a^4+a^2 c_1{}^2} \\ y(x)\to \sqrt {-a^4+a^2 c_1{}^2}+c_1 x \\ y(x)\to -\sqrt {a^2 x^2-a^4} \\ y(x)\to \sqrt {a^2 x^2-a^4} \\ \end{align*}