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]
\[ \boxed {-2 \sin \left (x \right ) y^{2}+3 y^{3}+\left (4 \cos \left (x \right ) y+9 x y^{2}\right ) y^{\prime }=2 x} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 638
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 {\frac {\left (972 x^{4}+36 \sqrt {3}\, \sqrt {\left (-x^{2}+c_{1} \right ) \left (-243 x^{4}+32 \cos \left (x \right )^{3}+243 c_{1} x^{2}\right )}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}{2}+\frac {8 \cos \left (x \right )^{2}}{\left (972 x^{4}+36 \sqrt {3}\, \sqrt {\left (-x^{2}+c_{1} \right ) \left (-243 x^{4}+32 \cos \left (x \right )^{3}+243 c_{1} x^{2}\right )}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}-2 \cos \left (x \right )}{9 x} \\ y \left (x \right ) &= -\frac {\left (\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}+4 \cos \left (x \right )\right ) \left (i \left (\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}-4 \cos \left (x \right )\right ) \sqrt {3}+\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}+4 \cos \left (x \right )\right )}{36 x \left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \left (\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}-4 \cos \left (x \right )\right ) \sqrt {3}-\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}-4 \cos \left (x \right )\right ) \left (\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}+4 \cos \left (x \right )\right )}{36 x \left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 34.362 (sec). Leaf size: 520
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]{-16 \cos ^3(x)+9 \left (27 x^4+27 c_1 x^2+\sqrt {3} \sqrt {x^2 \left (x^2+c_1\right ) \left (-32 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right )}\right )}+\frac {8 \cos ^2(x)}{\sqrt [3]{-8 \cos ^3(x)+\frac {9}{2} \left (27 x^4+27 c_1 x^2+\sqrt {3} \sqrt {x^2 \left (x^2+c_1\right ) \left (-32 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right )}\right )}}-4 \cos (x)}{18 x} \\ y(x)\to \frac {i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{-16 \cos ^3(x)+9 \left (27 x^4+27 c_1 x^2+\sqrt {3} \sqrt {x^2 \left (x^2+c_1\right ) \left (-32 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right )}\right )}-\frac {8 i \left (\sqrt {3}-i\right ) \cos ^2(x)}{\sqrt [3]{-8 \cos ^3(x)+\frac {9}{2} \left (27 x^4+27 c_1 x^2+\sqrt {3} \sqrt {x^2 \left (x^2+c_1\right ) \left (-32 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right )}\right )}}-8 \cos (x)}{36 x} \\ y(x)\to -\frac {2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-16 \cos ^3(x)+9 \left (27 x^4+27 c_1 x^2+\sqrt {3} \sqrt {x^2 \left (x^2+c_1\right ) \left (-32 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right )}\right )}-\frac {8 i \left (\sqrt {3}+i\right ) \cos ^2(x)}{\sqrt [3]{-8 \cos ^3(x)+\frac {9}{2} \left (27 x^4+27 c_1 x^2+\sqrt {3} \sqrt {x^2 \left (x^2+c_1\right ) \left (-32 \cos ^3(x)+243 x^2 \left (x^2+c_1\right )\right )}\right )}}+8 \cos (x)}{36 x} \\ \end{align*}