Internal problem ID [1036]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 2, First order equations. Exact equations. Section 2.5 Page 79
Problem number: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_exact]
\[ \boxed {-2 \sin \left (x \right ) y^{2}+3 y^{3}-2 x +\left (4 \cos \left (x \right ) y+9 x y^{2}\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 758
dsolve((-2*y(x)^2*sin(x)+3*y(x)^3-2*x)+(4*y(x)*cos(x)+9*x*y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}{18 x}+\frac {8 \cos \left (x \right )^{2}}{9 x \left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}-\frac {2 \cos \left (x \right )}{9 x} \\ y \left (x \right ) = -\frac {\left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}{36 x}-\frac {4 \cos \left (x \right )^{2}}{9 x \left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}-\frac {2 \cos \left (x \right )}{9 x}-\frac {i \sqrt {3}\, \left (\frac {\left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}{18 x}-\frac {8 \cos \left (x \right )^{2}}{9 x \left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \left (x \right ) = -\frac {\left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}{36 x}-\frac {4 \cos \left (x \right )^{2}}{9 x \left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}-\frac {2 \cos \left (x \right )}{9 x}+\frac {i \sqrt {3}\, \left (\frac {\left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}{18 x}-\frac {8 \cos \left (x \right )^{2}}{9 x \left (972 x^{4}+36 \sqrt {3}\, \sqrt {243 x^{6}-32 \cos \left (x \right )^{3} x^{2}-486 x^{4} c_{1} +32 \cos \left (x \right )^{3} c_{1} +243 c_{1}^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 33.84 (sec). Leaf size: 465
DSolve[(-2*y[x]^2*Sin[x]+3*y[x]^3-2*x)+(4*y[x]*Cos[x]+9*x*y[x]^2)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {2^{2/3} \sqrt [3]{243 x^2 \left (x^2+c_1\right )+9 \sqrt {729 x^4 \left (x^2+c_1\right ){}^2-96 x^2 \left (x^2+c_1\right ) \cos ^3(x)}-16 \cos ^3(x)}+\frac {8 \cos ^2(x)}{\sqrt [3]{-8 \cos ^3(x)+\frac {9}{2} \left (27 x^2 \left (x^2+c_1\right )+\sqrt {729 x^4 \left (x^2+c_1\right ){}^2-96 x^2 \left (x^2+c_1\right ) \cos ^3(x)}\right )}}-4 \cos (x)}{18 x} \\ y(x)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{243 x^2 \left (x^2+c_1\right )+9 \sqrt {729 x^4 \left (x^2+c_1\right ){}^2-96 x^2 \left (x^2+c_1\right ) \cos ^3(x)}-16 \cos ^3(x)}-\frac {16 \sqrt [3]{-2} \cos ^2(x)}{\sqrt [3]{243 x^2 \left (x^2+c_1\right )+9 \sqrt {729 x^4 \left (x^2+c_1\right ){}^2-96 x^2 \left (x^2+c_1\right ) \cos ^3(x)}-16 \cos ^3(x)}}-8 \cos (x)}{36 x} \\ y(x)\to -\frac {\cos (x) \left (8-\frac {16 (-1)^{2/3} \cos (x)}{\sqrt [3]{-8 \cos ^3(x)+\frac {9}{2} \left (27 x^2 \left (x^2+c_1\right )+\sqrt {729 x^4 \left (x^2+c_1\right ){}^2-96 x^2 \left (x^2+c_1\right ) \cos ^3(x)}\right )}}\right )+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{243 x^2 \left (x^2+c_1\right )+9 \sqrt {729 x^4 \left (x^2+c_1\right ){}^2-96 x^2 \left (x^2+c_1\right ) \cos ^3(x)}-16 \cos ^3(x)}}{36 x} \\ \end{align*}