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: 0.047 (sec). Leaf size: 420
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 = \frac {\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {2 c_{1}^{2}}{3 x \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}+\frac {c_{1}}{3 x} y = -\frac {\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {c_{1}^{2}}{3 x \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}+\frac {c_{1}}{3 x}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {2 c_{1}^{2}}{3 x \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}\right )}{2} y = -\frac {\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {c_{1}^{2}}{3 x \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}+\frac {c_{1}}{3 x}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {2 c_{1}^{2}}{3 x \left (-108 x^{4}+8 c_{1}^{3}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}\right )^{\frac {1}{3}}}\right )}{2} \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*}