32.4.10 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.2, page 90

Internal problem ID [5821]
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 10.2, page 90
Date solved : Monday, January 27, 2025 at 01:20:02 PM
CAS classification : [_exact]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 21

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

\[ \frac {x^{3}}{3}+y \left (x \right ) \sin \left (x \right )+\frac {y \left (x \right )^{4}}{4}+c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 60.196 (sec). Leaf size: 1119

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

\begin{align*} y(x)\to \frac {\sqrt {\frac {4 x^3+\left (27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {-\frac {8 \left (x^3-3 c_1\right )}{3 \sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}-\frac {2}{3} \sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}-\frac {4 \sqrt {6} \sin (x)}{\sqrt {\frac {4 x^3+\left (27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}}}} \\ y(x)\to \frac {\sqrt {\frac {4 x^3+\left (27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}}}{\sqrt {6}}+\frac {1}{2} \sqrt {-\frac {8 \left (x^3-3 c_1\right )}{3 \sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}-\frac {2}{3} \sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}-\frac {4 \sqrt {6} \sin (x)}{\sqrt {\frac {4 x^3+\left (27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}}}} \\ y(x)\to -\frac {\sqrt {\frac {4 x^3+\left (27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}}}{\sqrt {6}}-\frac {1}{2} \sqrt {-\frac {8 \left (x^3-3 c_1\right )}{3 \sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}-\frac {2}{3} \sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}+\frac {4 \sqrt {6} \sin (x)}{\sqrt {\frac {4 x^3+\left (27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}}}} \\ y(x)\to \frac {1}{2} \sqrt {-\frac {8 \left (x^3-3 c_1\right )}{3 \sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}-\frac {2}{3} \sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}+\frac {4 \sqrt {6} \sin (x)}{\sqrt {\frac {4 x^3+\left (27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}}}}-\frac {\sqrt {\frac {4 x^3+\left (27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}\right ){}^{2/3}-12 c_1}{\sqrt [3]{27 \sin ^2(x)+\sqrt {729 \sin ^4(x)-64 \left (x^3-3 c_1\right ){}^3}}}}}{\sqrt {6}} \\ \end{align*}