75.7.5 problem 179

Internal problem ID [16798]
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 : 179
Date solved : Tuesday, January 28, 2025 at 09:28:51 AM
CAS classification : [[_homogeneous, `class D`], _exact, _rational]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 65

dsolve((2*x+ (x^2+y(x)^2)/(x^2*y(x))  )=( (x^2+y(x)^2)/(x*y(x)^2) )*diff(y(x),x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.416 (sec). Leaf size: 78

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

\begin{align*} y(x)\to \frac {1}{2} x \left (x^2-\sqrt {x^4+2 c_1 x^2+4+c_1{}^2}+c_1\right ) \\ y(x)\to \frac {1}{2} x \left (x^2+\sqrt {x^4+2 c_1 x^2+4+c_1{}^2}+c_1\right ) \\ y(x)\to 0 \\ \end{align*}