32.4.15 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.7, page 90

Internal problem ID [5826]
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. Exercise 10.7, page 90
Date solved : Monday, January 27, 2025 at 01:20:07 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 25

dsolve((x^4*y(x)^2-y(x))+(x^2*y(x)^4-x)*diff(y(x),x)=0,y(x), singsol=all)
 

\[ -\frac {x^{3}}{3}-\frac {1}{y \left (x \right ) x}-\frac {y \left (x \right )^{3}}{3}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.153 (sec). Leaf size: 1507

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

\begin{align*} y(x)\to \frac {1}{4} \left (\sqrt {2} \sqrt {\frac {8 \sqrt [3]{2} x+2^{2/3} \left (x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}\right ){}^{2/3}}{x \sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}-2 \sqrt {-\frac {\sqrt [3]{x \left (x^4-3 c_1 x\right ){}^2+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}{\sqrt [3]{2} x}-\frac {2 \sqrt {2} \left (x^3-3 c_1\right )}{\sqrt {\frac {8 \sqrt [3]{2} x+2^{2/3} \left (x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}\right ){}^{2/3}}{x \sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}}-\frac {4 \sqrt [3]{2}}{\sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}\right ) \\ y(x)\to \frac {1}{4} \left (\sqrt {2} \sqrt {\frac {8 \sqrt [3]{2} x+2^{2/3} \left (x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}\right ){}^{2/3}}{x \sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}+2 \sqrt {-\frac {\sqrt [3]{x \left (x^4-3 c_1 x\right ){}^2+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}{\sqrt [3]{2} x}-\frac {2 \sqrt {2} \left (x^3-3 c_1\right )}{\sqrt {\frac {8 \sqrt [3]{2} x+2^{2/3} \left (x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}\right ){}^{2/3}}{x \sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}}-\frac {4 \sqrt [3]{2}}{\sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}\right ) \\ y(x)\to \frac {1}{4} \left (-\sqrt {2} \sqrt {\frac {8 \sqrt [3]{2} x+2^{2/3} \left (x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}\right ){}^{2/3}}{x \sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}-2 \sqrt {-\frac {\sqrt [3]{x \left (x^4-3 c_1 x\right ){}^2+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}{\sqrt [3]{2} x}+\frac {2 \sqrt {2} \left (x^3-3 c_1\right )}{\sqrt {\frac {8 \sqrt [3]{2} x+2^{2/3} \left (x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}\right ){}^{2/3}}{x \sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}}-\frac {4 \sqrt [3]{2}}{\sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}\right ) \\ y(x)\to \frac {1}{4} \left (2 \sqrt {-\frac {\sqrt [3]{x \left (x^4-3 c_1 x\right ){}^2+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}{\sqrt [3]{2} x}+\frac {2 \sqrt {2} \left (x^3-3 c_1\right )}{\sqrt {\frac {8 \sqrt [3]{2} x+2^{2/3} \left (x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}\right ){}^{2/3}}{x \sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}}-\frac {4 \sqrt [3]{2}}{\sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}-\sqrt {2} \sqrt {\frac {8 \sqrt [3]{2} x+2^{2/3} \left (x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}\right ){}^{2/3}}{x \sqrt [3]{x^9-6 c_1 x^6+9 c_1{}^2 x^3+\sqrt {x^2 \left (-256 x+\left (x^4-3 c_1 x\right ){}^4\right )}}}}\right ) \\ \end{align*}