8.6.12 problem 12

Internal problem ID [782]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Chapter 1 review problems. Page 78
Problem number : 12
Date solved : Monday, January 27, 2025 at 03:05:02 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 6 x y^{3}+2 y^{4}+\left (9 x^{2} y^{2}+8 x y^{3}\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 25

dsolve(6*x*y(x)^3+2*y(x)^4+(9*x^2*y(x)^2+8*x*y(x)^3)*diff(y(x),x) = 0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ 3 x^{2} y^{3}+2 x y^{4}+c_1 &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.152 (sec). Leaf size: 1714

DSolve[6*x*y[x]^3+2*y[x]^4+(9*x^2*y[x]^2+8*x*y[x]^3)*D[y[x],x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to 0 \\ y(x)\to \frac {1}{2} \sqrt {\frac {9 x^2}{16}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}}-\frac {1}{2} \sqrt {\frac {9 x^2}{8}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}-\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}-\frac {27 x^3}{32 \sqrt {\frac {9 x^2}{16}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}}}}-\frac {3 x}{8} \\ y(x)\to \frac {1}{2} \sqrt {\frac {9 x^2}{16}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}}+\frac {1}{2} \sqrt {\frac {9 x^2}{8}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}-\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}-\frac {27 x^3}{32 \sqrt {\frac {9 x^2}{16}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}}}}-\frac {3 x}{8} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {9 x^2}{16}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}}-\frac {1}{2} \sqrt {\frac {9 x^2}{8}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}-\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}+\frac {27 x^3}{32 \sqrt {\frac {9 x^2}{16}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}}}}-\frac {3 x}{8} \\ y(x)\to -\frac {1}{2} \sqrt {\frac {9 x^2}{16}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}}+\frac {1}{2} \sqrt {\frac {9 x^2}{8}+\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}-\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}+\frac {27 x^3}{32 \sqrt {\frac {9 x^2}{16}-\frac {4 \sqrt [3]{\frac {2}{3}} e^{c_1}}{\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {e^{2 c_1} x^3 \left (2187 x^5+2048 e^{c_1}\right )}-81 e^{c_1} x^4}}{2 \sqrt [3]{2} 3^{2/3} x}}}}-\frac {3 x}{8} \\ \end{align*}