32.3.1 problem Exact Differential equations. Exercise 9.4, page 79

Internal problem ID [5799]
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
Date solved : Monday, January 27, 2025 at 01:19:40 PM
CAS classification : [_exact, _rational]

\begin{align*} 3 x^{2} y+8 x y^{2}+\left (x^{3}+8 x^{2} y+12 y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 471

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 )^{{1}/{3}}}{6}+\frac {x^{3} \left (4 x -3\right )}{6 \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 )^{{1}/{3}}}-\frac {x^{2}}{3} \\ y \left (x \right ) &= \frac {\frac {\left (-1-i \sqrt {3}\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 )^{{2}/{3}}}{4}+x^{2} \left (-\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 )^{{1}/{3}}+x \left (i \sqrt {3}-1\right ) \left (x -\frac {3}{4}\right )\right )}{3 \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 )^{{1}/{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {1}{4}-\frac {i \sqrt {3}}{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 )^{{2}/{3}}+x^{2} \left (\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 )^{{1}/{3}}+x \left (1+i \sqrt {3}\right ) \left (x -\frac {3}{4}\right )\right )}{3 \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 )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.066 (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)*D[y[x],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*}