72.8.28 problem 43

Internal problem ID [14792]
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 : Tuesday, January 28, 2025 at 07:15:40 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 79

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

\[ y = \frac {-2 c_{1} {\mathrm e}^{-2 t} \left (\int {\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}d t \right )+2 i {\mathrm e}^{-2 t +\pi }+c_{1} {\mathrm e}^{t -\frac {3 \pi }{2}-\sin \left (t \right )}}{-c_{1} {\mathrm e}^{-2 t} \left (\int {\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}d t \right )+i {\mathrm e}^{-2 t +\pi }} \]

✓ Solution by Mathematica

Time used: 1.735 (sec). Leaf size: 254

DSolve[D[y[t],t]==(y[t]-2)*(y[t]+1-Cos[t]),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*}