Internal problem ID [13330]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page
90
Problem number: 4.7 (f).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {y^{\prime }-\frac {6 x^{2}+4}{3 y^{2}-4 y}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 418
dsolve(diff(y(x),x)=(6*x^2+4)/(3*y(x)^2-4*y(x)),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x +\frac {16}{27}\right ) \left (x^{3}+c_{1} +2 x \right )}\right )^{\frac {2}{3}}+2 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x +\frac {16}{27}\right ) \left (x^{3}+c_{1} +2 x \right )}\right )^{\frac {1}{3}}+4}{3 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x +\frac {16}{27}\right ) \left (x^{3}+c_{1} +2 x \right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x +\frac {16}{27}\right ) \left (x^{3}+c_{1} +2 x \right )}\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x +\frac {16}{27}\right ) \left (x^{3}+c_{1} +2 x \right )}\right )^{\frac {1}{3}}+4}{6 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x +\frac {16}{27}\right ) \left (x^{3}+c_{1} +2 x \right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x +\frac {16}{27}\right ) \left (x^{3}+c_{1} +2 x \right )}\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x +\frac {16}{27}\right ) \left (x^{3}+c_{1} +2 x \right )}\right )^{\frac {1}{3}}-4}{6 \left (8+27 x^{3}+27 c_{1} +54 x +3 \sqrt {81}\, \sqrt {\left (x^{3}+c_{1} +2 x +\frac {16}{27}\right ) \left (x^{3}+c_{1} +2 x \right )}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.967 (sec). Leaf size: 356
DSolve[y'[x]==(6*x^2+4)/(3*y[x]^2-4*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{6} \left (2^{2/3} \sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}+\frac {8 \sqrt [3]{2}}{\sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}}+4\right ) \\ y(x)\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}-\frac {8 \sqrt [3]{2} \left (1+i \sqrt {3}\right )}{\sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}}+8\right ) \\ y(x)\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}+\frac {8 i \sqrt [3]{2} \left (\sqrt {3}+i\right )}{\sqrt [3]{54 x^3+\sqrt {-256+\left (54 x^3+108 x+16+27 c_1\right ){}^2}+108 x+16+27 c_1}}+8\right ) \\ \end{align*}