Internal problem ID [574]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Miscellaneous problems, end of chapter 2. Page 133
Problem number: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {4 x^{3}+1}{y \left (2+3 y\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 660
dsolve(diff(y(x),x) = (4*x^3+1)/(y(x)*(2+3*y(x))),y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}{6}+\frac {2}{3 \left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{3} \\ y \relax (x ) = -\frac {\left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (-8+108 x^{4}+108 c_{1}+108 x +12 \sqrt {81 x^{8}+162 x^{5}+162 x^{4} c_{1}-12 x^{4}+81 x^{2}+162 x c_{1}+81 c_{1}^{2}-12 x -12 c_{1}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.432 (sec). Leaf size: 333
DSolve[y'[x]== (4*x^3+1)/(y[x]*(2+3*y[x])),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{27 x^4+\sqrt {-4+\left (27 \left (x^4+x\right )-2+27 c_1\right ){}^2}+27 x-2+27 c_1}+\frac {2 \sqrt [3]{2}}{\sqrt [3]{27 x^4+\sqrt {-4+\left (27 \left (x^4+x\right )-2+27 c_1\right ){}^2}+27 x-2+27 c_1}}-2\right ) \\ y(x)\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{27 x^4+\sqrt {-4+\left (27 \left (x^4+x\right )-2+27 c_1\right ){}^2}+27 x-2+27 c_1}-\frac {4 \sqrt [3]{-2}}{\sqrt [3]{27 x^4+\sqrt {-4+\left (27 \left (x^4+x\right )-2+27 c_1\right ){}^2}+27 x-2+27 c_1}}-4\right ) \\ y(x)\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{27 x^4+\sqrt {-4+\left (27 \left (x^4+x\right )-2+27 c_1\right ){}^2}+27 x-2+27 c_1}+\frac {4 (-1)^{2/3} \sqrt [3]{2}}{\sqrt [3]{27 x^4+\sqrt {-4+\left (27 \left (x^4+x\right )-2+27 c_1\right ){}^2}+27 x-2+27 c_1}}-4\right ) \\ \end{align*}