75.7.12 problem 187

Internal problem ID [16805]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 187
Date solved : Tuesday, January 28, 2025 at 09:31:28 AM
CAS classification : [_exact, _rational]

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

✓ Solution by Maple

Time used: 0.058 (sec). Leaf size: 129

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

\begin{align*} y &= \sqrt {-a^{2}-x^{2}-2 \sqrt {a^{2} x^{2}-c_{1}}} \\ y &= \sqrt {-a^{2}-x^{2}+2 \sqrt {a^{2} x^{2}-c_{1}}} \\ y &= -\sqrt {-a^{2}-x^{2}-2 \sqrt {a^{2} x^{2}-c_{1}}} \\ y &= -\sqrt {-a^{2}-x^{2}+2 \sqrt {a^{2} x^{2}-c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.400 (sec). Leaf size: 165

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

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