32.4.8 problem Recognizable Exact Differential equations. Integrating factors. Example 10.83, page 90

Internal problem ID [5819]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 10
Problem number : Recognizable Exact Differential equations. Integrating factors. Example 10.83, page 90
Date solved : Monday, January 27, 2025 at 01:19:58 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} y \left (2 x^{2} y^{3}+3\right )+x \left (x^{2} y^{3}-1\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.121 (sec). Leaf size: 39

dsolve((y(x)*(2*x^2*y(x)^3+3))+(x*(x^2*y(x)^3-1))*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {11 c_{1}}{3}} x^{3}}{\operatorname {RootOf}\left (11 \,{\mathrm e}^{11 c_{1}} \textit {\_Z}^{15}-{\mathrm e}^{11 c_{1}} \textit {\_Z}^{11}+4 x^{11}\right )^{5}} \]

✓ Solution by Mathematica

Time used: 11.496 (sec). Leaf size: 1081

DSolve[(y[x]*(2*x^2*y[x]^3+3))+(x*(x^2*y[x]^3-1))*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,1\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,2\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,3\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,4\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,5\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,6\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,7\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,8\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,9\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,10\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,11\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,12\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,13\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,14\right ] \\ y(x)\to \text {Root}\left [1024 \text {$\#$1}^{15} x^{22}+14080 \text {$\#$1}^{12} x^{20}+77440 \text {$\#$1}^9 x^{18}+212960 \text {$\#$1}^6 x^{16}-\text {$\#$1}^4 e^{\frac {44 c_1}{3}}+292820 \text {$\#$1}^3 x^{14}+161051 x^{12}\&,15\right ] \\ \end{align*}