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
:
Saturday, March 29, 2025 at 11:58:08 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*}
✓ Maple. Time used: 1.299 (sec). Leaf size: 128
ode:=diff(y(t),t) = 1+y(t)+y(t)^2*cos(t);
ic:=y(0) = 0;
dsolve([ode,ic],y(t), singsol=all);
\[
y = -\frac {4 \sec \left (t \right ) \operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right ) \left (\left (\cos \left (t \right )+\frac {\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )}{4}-\frac {1}{4}\right ) \operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+c_1 \left (\cos \left (t \right )+\frac {\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )}{4}-\frac {1}{4}\right ) \operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-\frac {\left (\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )-1\right ) \left (c_1 \operatorname {MathieuSPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+\operatorname {MathieuCPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )\right )}{4}\right )}{2 c_1 \operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-2 c_1 \operatorname {MathieuSPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+2 \operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-2 \operatorname {MathieuCPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )}
\]
✗ Mathematica
ode=D[y[t],t]== 1+y[t]+y[t]^2*Cos[t];
ic={y[0]==0};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
Not solved
✗ Sympy
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-y(t)**2*cos(t) - y(t) + Derivative(y(t), t) - 1,0)
ics = {y(0): 0}
dsolve(ode,func=y(t),ics=ics)
NotImplementedError : The given ODE -y(t)**2*cos(t) - y(t) + Derivative(y(t), t) - 1 cannot be solved by the lie group method