Internal problem ID [2047]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 12, page 46
Problem number: 15.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational]
\[ \boxed {2 y x +y^{4}+\left (x y^{3}-2 x^{2}\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 330
dsolve((2*x*y(x)+y(x)^4)+(x*y(x)^3-2*x^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\frac {\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}{2}+\frac {2 c_{1}^{2}}{\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}+c_{1}}{3 x} \\ y \left (x \right ) &= \frac {\left (-1-i \sqrt {3}\right ) \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {2}{3}}+4 c_{1} \left (i \sqrt {3}\, c_{1} -c_{1} +\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}\right )}{12 \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {2}{3}}+4 \left (-i \sqrt {3}\, c_{1} -c_{1} +\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}\right ) c_{1}}{12 \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}} x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 12.491 (sec). Leaf size: 371
DSolve[(2*x*y[x]+y[x]^4)+(x*y[x]^3-2*x^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\frac {2 \sqrt [3]{2} c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}+2 c_1}{6 x} \\ y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}-4 c_1}{12 x} \\ y(x)\to \frac {\frac {2 \sqrt [3]{2} \left (1-i \sqrt {3}\right ) c_1{}^2}{\sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^4+3 \sqrt {81 x^8+12 c_1{}^3 x^4}+2 c_1{}^3}-4 c_1}{12 x} \\ y(x)\to 0 \\ \end{align*}