Internal problem ID [4455]
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]
\[ \boxed {3 x^{2} y+8 y^{2} x +\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime }=0} \]
✓ 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 \left (x \right ) = \frac {\left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+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 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}-\frac {x^{2}}{3} y \left (x \right ) = -\frac {\left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+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 c_{1} x^{6}-54 c_{1} x^{5}+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 c_{1} x^{6}-54 c_{1} x^{5}+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 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} y \left (x \right ) = -\frac {\left (9 x^{5}-27 c_{1} -8 x^{6}+3 \sqrt {-3 x^{10}+3 x^{9}+48 c_{1} x^{6}-54 c_{1} x^{5}+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 c_{1} x^{6}-54 c_{1} x^{5}+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 c_{1} x^{6}-54 c_{1} x^{5}+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 c_{1} x^{6}-54 c_{1} x^{5}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \end{align*}
✓ Solution by Mathematica
Time used: 1.703 (sec). Leaf size: 474
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]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}+\frac {(4 x-3) x^3}{\sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}}\right ) y(x)\to \frac {1}{48} \left (-16 x^2+4 i \left (\sqrt {3}+i\right ) \sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}-\frac {4 i \left (\sqrt {3}-i\right ) (4 x-3) x^3}{\sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}}\right ) y(x)\to \frac {1}{48} \left (-16 x^2-4 \left (1+i \sqrt {3}\right ) \sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}+\frac {4 i \left (\sqrt {3}+i\right ) (4 x-3) x^3}{\sqrt [3]{-8 x^6+9 x^5+3 \sqrt {3} \sqrt {-x^{10}+x^9-16 c_1 x^6+18 c_1 x^5+27 c_1{}^2}+27 c_1}}\right ) \end{align*}