13.5.2 problem 5
Internal
problem
ID
[2348]
Book
:
Differential
equations
and
their
applications,
3rd
ed.,
M.
Braun
Section
:
Section
1.10.
Page
80
Problem
number
:
5
Date
solved
:
Tuesday, January 28, 2025 at 02:35:56 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=1+y+y^{2} \cos \left (t \right ) \end{align*}
With initial conditions
\begin{align*} y \left (0\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 1.635 (sec). Leaf size: 128
dsolve([diff(y(t),t)= 1+y(t)+y(t)^2*cos(t),y(0) = 0],y(t), singsol=all)
\[
y = -\frac {4 \sec \left (t \right ) \operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right ) \left (\left (\cos \left (t \right )+\frac {\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )}{4}-\frac {1}{4}\right ) \operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+c_1 \left (\cos \left (t \right )+\frac {\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )}{4}-\frac {1}{4}\right ) \operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-\frac {\left (\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )-1\right ) \left (c_1 \operatorname {MathieuSPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+\operatorname {MathieuCPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )\right )}{4}\right )}{2 c_1 \operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-2 c_1 \operatorname {MathieuSPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+2 \operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-2 \operatorname {MathieuCPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )}
\]
✗ Solution by Mathematica
Time used: 0.000 (sec). Leaf size: 0
DSolve[{D[y[t],t]== 1+y[t]+y[t]^2*Cos[t],{y[0]==0}},y[t],t,IncludeSingularSolutions -> True]
Not solved