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]
\[ \boxed {y^{\prime }-\frac {4 x^{3}+1}{y \left (2+3 y\right )}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 382
dsolve(diff(y(x),x) = (4*x^3+1)/(y(x)*(2+3*y(x))),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {2}{3}}-2 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}+4}{6 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}+4}{12 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}-4}{12 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.502 (sec). Leaf size: 356
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 x^4+27 x-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 x^4+27 x-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 x^4+27 x-2+27 c_1\right ){}^2}+27 x-2+27 c_1}-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{27 x^4+\sqrt {-4+\left (27 x^4+27 x-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 x^4+27 x-2+27 c_1\right ){}^2}+27 x-2+27 c_1}+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right )}{\sqrt [3]{27 x^4+\sqrt {-4+\left (27 x^4+27 x-2+27 c_1\right ){}^2}+27 x-2+27 c_1}}-4\right ) \\ \end{align*}