83.8.9 problem 10

Internal problem ID [19135]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Misc examples on chapter II at page 25
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 01:05:03 PM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 257

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

\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-4 b^{2}+4 x^{2}-2 \sqrt {8 x^{4}+\left (-8 a^{2}-8 b^{2}\right ) x^{2}+2 a^{4}+4 a^{2} b^{2}+2 b^{4}+16 c_{1}}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-4 b^{2}+4 x^{2}-2 \sqrt {8 x^{4}+\left (-8 a^{2}-8 b^{2}\right ) x^{2}+2 a^{4}+4 a^{2} b^{2}+2 b^{4}+16 c_{1}}}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {-4 b^{2}+4 x^{2}+2 \sqrt {8 x^{4}+\left (-8 a^{2}-8 b^{2}\right ) x^{2}+2 a^{4}+4 a^{2} b^{2}+2 b^{4}+16 c_{1}}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-4 b^{2}+4 x^{2}+2 \sqrt {8 x^{4}+\left (-8 a^{2}-8 b^{2}\right ) x^{2}+2 a^{4}+4 a^{2} b^{2}+2 b^{4}+16 c_{1}}}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.632 (sec). Leaf size: 209

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

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