72.8.28 problem 43
Internal
problem
ID
[14713]
Book
:
DIFFERENTIAL
EQUATIONS
by
Paul
Blanchard,
Robert
L.
Devaney,
Glen
R.
Hall.
4th
edition.
Brooks/Cole.
Boston,
USA.
2012
Section
:
Chapter
1.
First-Order
Differential
Equations.
Review
Exercises
for
chapter
1.
page
136
Problem
number
:
43
Date
solved
:
Thursday, March 13, 2025 at 04:16:08 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\left (y-2\right ) \left (y+1-\cos \left (t \right )\right ) \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 79
ode:=diff(y(t),t) = (y(t)-2)*(y(t)+1-cos(t));
dsolve(ode,y(t), singsol=all);
\[
y = \frac {-2 c_{1} {\mathrm e}^{-2 t} \left (\int {\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}d t \right )+2 i {\mathrm e}^{-2 t +\pi }+c_{1} {\mathrm e}^{t -\frac {3 \pi }{2}-\sin \left (t \right )}}{-c_{1} {\mathrm e}^{-2 t} \left (\int {\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}d t \right )+i {\mathrm e}^{-2 t +\pi }}
\]
✓ Mathematica. Time used: 1.735 (sec). Leaf size: 254
ode=D[y[t],t]==(y[t]-2)*(y[t]+1-Cos[t]);
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*}
y(t)\to -\frac {-2 \int _1^{e^{i t}}e^{\frac {i \left (K[1]^2-(3-i) K[1]-1\right )}{2 K[1]}} K[1]^{-1-3 i}dK[1]+i \left (e^{i t}\right )^{-3 i} \exp \left (\frac {1}{2} \left (-i e^{-i t}+i e^{i t}+(-1-3 i)\right )\right )-2 c_1}{\int _1^{e^{i t}}e^{\frac {i \left (K[1]^2-(3-i) K[1]-1\right )}{2 K[1]}} K[1]^{-1-3 i}dK[1]+c_1} \\
y(t)\to 2 \\
y(t)\to 2-\frac {i \left (e^{i t}\right )^{-3 i} \exp \left (\frac {1}{2} \left (-i e^{-i t}+i e^{i t}+(-1-3 i)\right )\right )}{\int _1^{e^{i t}}\exp \left (\frac {1}{2} i \left (K[1]-(6-2 i) \log (K[1])-(3-i)-\frac {1}{K[1]}\right )\right )dK[1]} \\
\end{align*}
✓ Sympy. Time used: 22.278 (sec). Leaf size: 53
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-(y(t) - 2)*(y(t) - cos(t) + 1) + Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = \frac {\left (2 C_{1} e^{\sin {\left (t \right )}} + e^{3 t} - 2 e^{\sin {\left (t \right )}} \int e^{3 t} e^{- \sin {\left (t \right )}}\, dt\right ) e^{- \sin {\left (t \right )}}}{C_{1} - \int e^{3 t} e^{- \sin {\left (t \right )}}\, dt}
\]