Internal problem ID [12474]
Book: DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR
PUBLISHERS, Moscow 1969.
Section: Chapter 8. Differential equations. Exercises page 595
Problem number: 76.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
\[ \boxed {6 y^{2} x +3 \left (2 x^{2} y+y^{2}\right ) y^{\prime }=-4 x^{3}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 422
dsolve(2*(3*x*y(x)^2+2*x^3)+3*(2*x^2*y(x)+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {4 x^{10}+x^{8}+4 c_{1} x^{6}+2 c_{1} x^{4}+c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{4}}{\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {4 x^{10}+x^{8}+4 c_{1} x^{6}+2 c_{1} x^{4}+c_{1}^{2}}\right )^{\frac {1}{3}}}-x^{2} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {2}{3}}-4 x^{4}-4 x^{2} \left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}-\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {2}{3}}}{4 \left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}}{4}-\frac {\left (i \sqrt {3}\, x^{2}+x^{2}+\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}\right ) x^{2}}{\left (-4 x^{4}-4 c_{1} -8 x^{6}+4 \sqrt {\left (4 x^{6}+x^{4}+c_{1} \right ) \left (x^{4}+c_{1} \right )}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 25.227 (sec). Leaf size: 419
DSolve[2*(3*x*y[x]^2+2*x^3)+3*(2*x^2*y[x]+y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x^2+\frac {\sqrt [3]{2} x^4}{\sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}}+\frac {\sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}}{\sqrt [3]{2}} \\ y(x)\to \frac {1}{4} \left (-4 x^2-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^4}{\sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}\right ) \\ y(x)\to \frac {1}{4} \left (-4 x^2+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}}+2^{2/3} \left (-1-i \sqrt {3}\right ) \sqrt [3]{-2 x^6-x^4+\sqrt {4 x^{10}+x^8-4 c_1 x^6-2 c_1 x^4+c_1{}^2}+c_1}\right ) \\ \end{align*}