Internal problem ID [522]
Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and
DiPrima
Section: Section 2.4. Page 76
Problem number: 11.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {t^{2}+1}{3 y-y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 660
dsolve(diff(y(t),t) = (t^2+1)/(3*y(t)-y(t)^2),y(t), singsol=all)
\begin{align*} y \relax (t ) = \frac {\left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}{2}+\frac {9}{2 \left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}+\frac {3}{2} \\ y \relax (t ) = -\frac {\left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}{4}-\frac {9}{4 \left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}+\frac {3}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}{2}-\frac {9}{2 \left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (t ) = -\frac {\left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}{4}-\frac {9}{4 \left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}+\frac {3}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}{2}-\frac {9}{2 \left (27-4 t^{3}-12 c_{1}-12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1}-162 t}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 3.159 (sec). Leaf size: 342
DSolve[y'[t] == (t^2+1)/(3*y[t]-y[t]^2),y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{2} \left (\sqrt [3]{-4 t^3+\sqrt {-729+\left (4 t \left (t^2+3\right )-3 (9+4 c_1)\right ){}^2}-12 t+27+12 c_1}+\frac {9}{\sqrt [3]{-4 t^3+\sqrt {-729+\left (4 t \left (t^2+3\right )-3 (9+4 c_1)\right ){}^2}-12 t+27+12 c_1}}+3\right ) \\ y(t)\to \frac {1}{4} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{-4 t^3+\sqrt {-729+\left (4 t \left (t^2+3\right )-3 (9+4 c_1)\right ){}^2}-12 t+27+12 c_1}+\frac {-9-9 i \sqrt {3}}{\sqrt [3]{-4 t^3+\sqrt {-729+\left (4 t \left (t^2+3\right )-3 (9+4 c_1)\right ){}^2}-12 t+27+12 c_1}}+6\right ) \\ y(t)\to \frac {1}{4} \left (-\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-4 t^3+\sqrt {-729+\left (4 t \left (t^2+3\right )-3 (9+4 c_1)\right ){}^2}-12 t+27+12 c_1}\right )+\frac {9 i \left (\sqrt {3}+i\right )}{\sqrt [3]{-4 t^3+\sqrt {-729+\left (4 t \left (t^2+3\right )-3 (9+4 c_1)\right ){}^2}-12 t+27+12 c_1}}+6\right ) \\ \end{align*}