Internal problem ID [6089]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 5. Existence and uniqueness of solutions to first order equations. Page
198
Problem number: 2(c).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {5 x^{3} y^{2}+2 y+\left (3 y x^{4}+2 x \right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.515 (sec). Leaf size: 350
dsolve((5*x^3*y(x)^2+2*y(x))+(3*x^4*y(x)+2*x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\frac {12^{\frac {2}{3}} \left (12^{\frac {1}{3}} c_{1}^{2}+{\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}}\right )^{2}}{36 c_{1}^{2} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}}}-1}{x^{3}} \\ y \left (x \right ) &= \frac {-\frac {c_{1} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}}}{3}+\frac {3 \,2^{\frac {1}{3}} \left (x^{2}+\frac {\sqrt {-12 c_{1}^{4}+81 x^{4}}}{9}\right ) \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {1}{3}}}{4}-\frac {\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} c_{1}^{3}}{6}}{c_{1} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}} x^{3}} \\ y \left (x \right ) &= -\frac {3 \left (\frac {4 c_{1} {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}}}{9}+2^{\frac {1}{3}} \left (x^{2}+\frac {\sqrt {-12 c_{1}^{4}+81 x^{4}}}{9}\right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {1}{3}}-\frac {2 \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} c_{1}^{3}}{9}\right )}{4 {\left (\left (9 x^{2}+\sqrt {-12 c_{1}^{4}+81 x^{4}}\right ) c_{1} \right )}^{\frac {2}{3}} x^{3} c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 49.208 (sec). Leaf size: 400
DSolve[(5*x^3*y[x]^2+2*y[x])+(3*x^4*y[x]+2*x)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 x^2+\frac {2 x^4}{\sqrt [3]{\frac {27 c_1 x^{10}}{2}-x^6+\frac {3}{2} \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}+2^{2/3} \sqrt [3]{27 c_1 x^{10}-2 x^6+3 \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}{6 x^5} \\ y(x)\to \frac {-4 x^2-\frac {2 \left (1+i \sqrt {3}\right ) x^4}{\sqrt [3]{\frac {27 c_1 x^{10}}{2}-x^6+\frac {3}{2} \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}+i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{27 c_1 x^{10}-2 x^6+3 \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}{12 x^5} \\ y(x)\to -\frac {4 x^2-\frac {2 i \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{\frac {27 c_1 x^{10}}{2}-x^6+\frac {3}{2} \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1 x^{10}-2 x^6+3 \sqrt {3} \sqrt {c_1 x^{16} \left (-4+27 c_1 x^4\right )}}}{12 x^5} \\ \end{align*}