4.3.7 problem 11
Internal
problem
ID
[1172]
Book
:
Elementary
differential
equations
and
boundary
value
problems,
10th
ed.,
Boyce
and
DiPrima
Section
:
Section
2.4.
Page
76
Problem
number
:
11
Date
solved
:
Tuesday, September 30, 2025 at 04:27:17 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {t^{2}+1}{3 y-y^{2}} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 440
ode:=diff(y(t),t) = (t^2+1)/(3*y(t)-y(t)^2);
dsolve(ode,y(t), singsol=all);
\begin{align*}
y &= \frac {\left (27-4 t^{3}-12 c_1 -12 t +2 \sqrt {4 t^{6}+24 t^{3} c_1 +24 t^{4}-54 t^{3}+36 c_1^{2}+72 c_1 t +36 t^{2}-162 c_1 -162 t}\right )^{{1}/{3}}}{2}+\frac {9}{2 \left (27-4 t^{3}-12 c_1 -12 t +2 \sqrt {4 t^{6}+24 t^{3} c_1 +24 t^{4}-54 t^{3}+36 c_1^{2}+72 c_1 t +36 t^{2}-162 c_1 -162 t}\right )^{{1}/{3}}}+\frac {3}{2} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (27-4 t^{3}-12 c_1 -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 c_1 +3 t \right ) \left (t^{3}+3 t +3 c_1 -\frac {27}{2}\right )}\right )^{{2}/{3}}-9 i \sqrt {3}-6 \left (27-4 t^{3}-12 c_1 -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 c_1 +3 t \right ) \left (t^{3}+3 t +3 c_1 -\frac {27}{2}\right )}\right )^{{1}/{3}}+9}{4 \left (27-4 t^{3}-12 c_1 -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 c_1 +3 t \right ) \left (t^{3}+3 t +3 c_1 -\frac {27}{2}\right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (27-4 t^{3}-12 c_1 -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 c_1 +3 t \right ) \left (t^{3}+3 t +3 c_1 -\frac {27}{2}\right )}\right )^{{2}/{3}}-9 i \sqrt {3}+6 \left (27-4 t^{3}-12 c_1 -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 c_1 +3 t \right ) \left (t^{3}+3 t +3 c_1 -\frac {27}{2}\right )}\right )^{{1}/{3}}-9}{4 \left (27-4 t^{3}-12 c_1 -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 c_1 +3 t \right ) \left (t^{3}+3 t +3 c_1 -\frac {27}{2}\right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 2.365 (sec). Leaf size: 343
ode=D[y[t],t] == (t^2+1)/(3*y[t]-y[t]^2);
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \frac {1}{2} \left (\sqrt [3]{-4 t^3+\sqrt {-729+\left (4 t^3+12 t-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^3+12 t-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^3+12 t-3 (9+4 c_1)\right ){}^2}-12 t+27+12 c_1}-\frac {9 \left (1+i \sqrt {3}\right )}{\sqrt [3]{-4 t^3+\sqrt {-729+\left (4 t^3+12 t-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^3+12 t-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^3+12 t-3 (9+4 c_1)\right ){}^2}-12 t+27+12 c_1}}+6\right ) \end{align*}
✓ Sympy. Time used: 94.783 (sec). Leaf size: 634
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq((-t**2 - 1)/(-y(t)**2 + 3*y(t)) + Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
\text {Solution too large to show}
\]