Internal problem ID [586]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Miscellaneous problems, end of chapter 2. Page 133
Problem number: 19.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {3 x^{2}-2 y-y^{3}}{2 x +3 x y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 507
dsolve(diff(y(x),x) = (3*x^2-2*y(x)-y(x)^3)/(2*x+3*x*y(x)^2),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {4 x}{\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}+\frac {2 x}{\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {4 x}{\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}+\frac {2 x}{\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {4 x}{\left (\left (108 x^{3}+12 \sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.344 (sec). Leaf size: 358
DSolve[y'[x] == (3*x^2-2*y[x]-y[x]^3)/(2*x+3*x*y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}{3 \sqrt [3]{2} x}-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}} \\ y(x)\to \frac {\sqrt [3]{2} \left (1+i \sqrt {3}\right ) x}{\sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}{6 \sqrt [3]{2} x} \\ y(x)\to \frac {\sqrt [3]{2} \left (1-i \sqrt {3}\right ) x}{\sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^5+27 c_1 x^2+\sqrt {864 x^6+729 x^4 \left (x^3+c_1\right ){}^2}}}{6 \sqrt [3]{2} x} \\ \end{align*}