Internal problem ID [5438]
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)]]]
Solve \begin {gather*} \boxed {2 y^{3} x +\cos \relax (x ) y+\left (3 y^{2} x^{2}+\sin \relax (x )\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 375
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 \relax (x ) = \frac {\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}}{6 x}-\frac {2 \sin \relax (x )}{x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}}{12 x}+\frac {\sin \relax (x )}{x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}}{6 x}+\frac {2 \sin \relax (x )}{x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}}{12 x}+\frac {\sin \relax (x )}{x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}}{6 x}+\frac {2 \sin \relax (x )}{x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \left (\sin ^{3}\relax (x )\right )}-108 c_{1} x \right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 30.809 (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*}