Internal problem ID [3947]
Book: Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section: Chapter 2. Special types of differential equations of the first kind. Lesson 9
Problem number: Exact Differential equations. Exercise 9.4, page 79.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, _rational]
Solve \begin {gather*} \boxed {3 x^{2} y+8 x y^{2}+\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 597
dsolve((3*x^2*y(x)+8*x*y(x)^2)+(x^3+8*x^2*y(x)+12*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {1}{12} x^{3}-\frac {1}{9} x^{4}\right )}{\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x^{2}}{3} \\ y \relax (x ) = -\frac {\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {\frac {1}{4} x^{3}-\frac {1}{3} x^{4}}{\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x^{2}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {\frac {1}{2} x^{3}-\frac {2}{3} x^{4}}{\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}+\frac {\frac {1}{4} x^{3}-\frac {1}{3} x^{4}}{\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x^{2}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {\frac {1}{2} x^{3}-\frac {2}{3} x^{4}}{\left (9 x^{5}-27 c_{1}-8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 x^{6} c_{1}-54 x^{5} c_{1}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.643 (sec). Leaf size: 431
DSolve[(3*x^2*y[x]+8*x*y[x]^2)+(x^3+8*x^2*y[x]+12*y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (-2 x^2+\sqrt [3]{(9-8 x) x^5+3 \left (\sqrt {-3 (x-1) x^9+6 c_1 (9-8 x) x^5+81 c_1{}^2}+9 c_1\right )}+\frac {(4 x-3) x^3}{\sqrt [3]{(9-8 x) x^5+3 \left (\sqrt {-3 (x-1) x^9+6 c_1 (9-8 x) x^5+81 c_1{}^2}+9 c_1\right )}}\right ) \\ y(x)\to \frac {1}{48} \left (-16 x^2+4 i \left (\sqrt {3}+i\right ) \sqrt [3]{(9-8 x) x^5+3 \left (\sqrt {-3 (x-1) x^9+6 c_1 (9-8 x) x^5+81 c_1{}^2}+9 c_1\right )}+\frac {\left (-4-4 i \sqrt {3}\right ) (4 x-3) x^3}{\sqrt [3]{(9-8 x) x^5+3 \left (\sqrt {-3 (x-1) x^9+6 c_1 (9-8 x) x^5+81 c_1{}^2}+9 c_1\right )}}\right ) \\ y(x)\to \frac {1}{48} \left (-16 x^2-4 \left (1+i \sqrt {3}\right ) \sqrt [3]{(9-8 x) x^5+3 \left (\sqrt {-3 (x-1) x^9+6 c_1 (9-8 x) x^5+81 c_1{}^2}+9 c_1\right )}+\frac {4 i \left (\sqrt {3}+i\right ) (4 x-3) x^3}{\sqrt [3]{(9-8 x) x^5+3 \left (\sqrt {-3 (x-1) x^9+6 c_1 (9-8 x) x^5+81 c_1{}^2}+9 c_1\right )}}\right ) \\ \end{align*}