12.7.11 problem 11

Internal problem ID [1721]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Exact equations. Integrating factors. Section 2.6 Page 91
Problem number : 11
Date solved : Monday, January 27, 2025 at 05:32:31 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} 12 x^{3} y+24 x^{2} y^{2}+\left (9 x^{4}+32 x^{3} y+4 y\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 27

dsolve((12*x^3*y(x)+24*x^2*y(x)^2)+(9*x^4+32*x^3*y(x)+4*y(x))*diff(y(x),x)=0,y(x), singsol=all)
 

\[ 3 x^{4} y^{3}+8 x^{3} y^{4}+y^{4}+c_1 = 0 \]

✓ Solution by Mathematica

Time used: 61.712 (sec). Leaf size: 1733

DSolve[(12*x^3*y[x]+24*x^2*y[x]^2)+(9*x^4+32*x^3*y[x]+4*y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {3 x^4}{32 x^3+4}+\frac {1}{2} \sqrt {\frac {9 x^8}{4 \left (8 x^3+1\right )^2}+\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}-\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}}-\frac {1}{2} \sqrt {\frac {9 x^8}{2 \left (8 x^3+1\right )^2}-\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}+\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}-\frac {27 x^{12}}{4 \left (8 x^3+1\right )^3 \sqrt {\frac {9 x^8}{4 \left (8 x^3+1\right )^2}+\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}-\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}}}} \\ y(x)\to -\frac {3 x^4}{32 x^3+4}+\frac {1}{2} \sqrt {\frac {9 x^8}{4 \left (8 x^3+1\right )^2}+\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}-\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}}+\frac {1}{2} \sqrt {\frac {9 x^8}{2 \left (8 x^3+1\right )^2}-\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}+\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}-\frac {27 x^{12}}{4 \left (8 x^3+1\right )^3 \sqrt {\frac {9 x^8}{4 \left (8 x^3+1\right )^2}+\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}-\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}}}} \\ y(x)\to -\frac {3 x^4}{32 x^3+4}-\frac {1}{2} \sqrt {\frac {9 x^8}{4 \left (8 x^3+1\right )^2}+\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}-\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}}-\frac {1}{2} \sqrt {\frac {9 x^8}{2 \left (8 x^3+1\right )^2}-\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}+\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}+\frac {27 x^{12}}{4 \left (8 x^3+1\right )^3 \sqrt {\frac {9 x^8}{4 \left (8 x^3+1\right )^2}+\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}-\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}}}} \\ y(x)\to -\frac {3 x^4}{32 x^3+4}-\frac {1}{2} \sqrt {\frac {9 x^8}{4 \left (8 x^3+1\right )^2}+\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}-\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}}+\frac {1}{2} \sqrt {\frac {9 x^8}{2 \left (8 x^3+1\right )^2}-\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}+\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}+\frac {27 x^{12}}{4 \left (8 x^3+1\right )^3 \sqrt {\frac {9 x^8}{4 \left (8 x^3+1\right )^2}+\frac {\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}{3 \sqrt [3]{2} \left (8 x^3+1\right )}-\frac {4 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {59049 c_1{}^2 x^{16}+6912 c_1{}^3 \left (8 x^3+1\right )^3}-243 c_1 x^8}}}}} \\ \end{align*}