Internal problem ID [5336]
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]]
Solve \begin {gather*} \boxed {5 y^{2} x^{3}+2 y+\left (3 x^{4} y+2 x \right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.828 (sec). Leaf size: 347
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 \relax (x ) = \frac {\frac {\left (\frac {6 \left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}{c_{1}}+\frac {72 c_{1}}{\left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}\right )^{2}}{1296}-1}{x^{3}} \\ y \relax (x ) = \frac {\frac {\left (-\frac {3 \left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}{c_{1}}-\frac {36 c_{1}}{\left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}-18 i \sqrt {3}\, \left (\frac {\left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2 c_{1}}{\left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}\right )\right )^{2}}{1296}-1}{x^{3}} \\ y \relax (x ) = \frac {\frac {\left (-\frac {3 \left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}{c_{1}}-\frac {36 c_{1}}{\left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}+18 i \sqrt {3}\, \left (\frac {\left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}{6 c_{1}}-\frac {2 c_{1}}{\left (\left (108 x^{2}+12 \sqrt {81 x^{4}-12 c_{1}^{4}}\right ) c_{1}\right )^{\frac {1}{3}}}\right )\right )^{2}}{1296}-1}{x^{3}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 58.101 (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*}