Internal problem ID [3969]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
Solve \begin {gather*} \boxed {x^{2}+\cos \relax (x ) y+\left (y^{3}+\sin \relax (x )\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (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)
\[ \frac {x^{3}}{3}+y \relax (x ) \sin \relax (x )+\frac {y \relax (x )^{4}}{4}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 60.2 (sec). Leaf size: 1119
DSolve[(x^2+y[x]*Cos[x])+(y[x]^3+Sin[x])*y'[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*}