82.12.11 problem Ex. 11

Internal problem ID [18776]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 11
Date solved : Tuesday, January 28, 2025 at 12:16:50 PM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 257

dsolve(x*(x^2+y(x)^2-a^2)+y(x)*(x^2-y(x)^2-b^2)*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: 2.633 (sec). Leaf size: 209

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