72.8.28 problem 43

Internal problem ID [14721]
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 : Monday, March 31, 2025 at 12:55:22 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=\left (y-2\right ) \left (y+1-\cos \left (t \right )\right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 65

ode:=diff(y(t),t) = (y(t)-2)*(y(t)+1-cos(t)); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {2 i {\mathrm e}^{\pi }+c_1 \,{\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}-2 \int {\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}d t c_1}{i {\mathrm e}^{\pi }-\int {\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}d t c_1} \]

✓ 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)
 

\[ y{\left (t \right )} = \frac {\left (2 C_{1} e^{\sin {\left (t \right )}} + e^{3 t} - 2 e^{\sin {\left (t \right )}} \int e^{3 t} e^{- \sin {\left (t \right )}}\, dt\right ) e^{- \sin {\left (t \right )}}}{C_{1} - \int e^{3 t} e^{- \sin {\left (t \right )}}\, dt} \]