Internal problem ID [6191]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.5. Exact Equations. Page
20
Problem number: 10.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\[ \boxed {2 y^{3} x +y \cos \left (x \right )+\left (3 y^{2} x^{2}+\sin \left (x \right )\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 302
dsolve((2*x*y(x)^3+y(x)*cos(x))+(3*x^2*y(x)^2+sin(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}-12 \sin \left (x \right )}{6 x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {i \sqrt {3}\, \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}+12 i \sin \left (x \right ) \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}-12 \sin \left (x \right )}{12 x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}+12 i \sin \left (x \right ) \sqrt {3}-\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}+12 \sin \left (x \right )}{12 x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 34.571 (sec). Leaf size: 339
DSolve[(2*x*y[x]^3+y[x]*Cos[x])+(3*x^2*y[x]^2+Sin[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{9 c_1 x^4+\sqrt {12 x^6 \sin ^3(x)+81 c_1{}^2 x^8}}}{\sqrt [3]{2} 3^{2/3} x^2}-\frac {\sqrt [3]{\frac {2}{3}} \sin (x)}{\sqrt [3]{9 c_1 x^4+\sqrt {12 x^6 \sin ^3(x)+81 c_1{}^2 x^8}}} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sin (x)}{2^{2/3} \sqrt [3]{27 c_1 x^4+3 \sqrt {12 x^6 \sin ^3(x)+81 c_1{}^2 x^8}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 c_1 x^4+\sqrt {108 x^6 \sin ^3(x)+729 c_1{}^2 x^8}}}{6 \sqrt [3]{2} x^2} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sin (x)}{2^{2/3} \sqrt [3]{27 c_1 x^4+3 \sqrt {12 x^6 \sin ^3(x)+81 c_1{}^2 x^8}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1 x^4+\sqrt {108 x^6 \sin ^3(x)+729 c_1{}^2 x^8}}}{6 \sqrt [3]{2} x^2} \\ \end{align*}