Internal problem ID [12736]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\left (-2+y\right ) \left (y+1-\cos \left (t \right )\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 118
dsolve(diff(y(t),t)=(y(t)-2)*(y(t)+1-cos(t)),y(t), singsol=all)
\[ y \left (t \right ) = -\frac {i c_{1} {\mathrm e}^{t -\frac {3 \pi }{2}-\sin \left (t \right )}}{c_{1} {\mathrm e}^{-2 t} \left (\int i {\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}d t \right )+{\mathrm e}^{\pi -2 t}}-\frac {-2 c_{1} {\mathrm e}^{-2 t} \left (\int i {\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}d t \right )-2 \,{\mathrm e}^{\pi -2 t}}{c_{1} {\mathrm e}^{-2 t} \left (\int i {\mathrm e}^{-\frac {3 \pi }{2}+3 t -\sin \left (t \right )}d t \right )+{\mathrm e}^{\pi -2 t}} \]
✓ Solution by Mathematica
Time used: 3.379 (sec). Leaf size: 224
DSolve[y'[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]^2-1\right )}{2 K[1]}} K[1]^{-1-3 i}dK[1]+i e^{\frac {1}{2} i e^{-i t} \left (-1+e^{2 i t}\right )} \left (e^{i t}\right )^{-3 i}-2 c_1}{\int _1^{e^{i t}}e^{\frac {i \left (K[1]^2-1\right )}{2 K[1]}} K[1]^{-1-3 i}dK[1]+c_1} y(t)\to 2 y(t)\to 2-\frac {i e^{\frac {1}{2} i e^{-i t} \left (-1+e^{2 i t}\right )} \left (e^{i t}\right )^{-3 i}}{\int _1^{e^{i t}}e^{\frac {i \left (K[1]^2-1\right )}{2 K[1]}} K[1]^{-1-3 i}dK[1]} \end{align*}