14.4.5 problem 5

Internal problem ID [2523]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.10. Existence-uniqueness theorem. Excercises page 80
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:08:09 AM
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: 1.337 (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);
 

\[ y = -\frac {4 \,\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right ) \sec \left (t \right ) \left (\left (\cos \left (t \right )+\frac {\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )}{4}-\frac {1}{4}\right ) \operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+c_1 \left (\cos \left (t \right )+\frac {\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )}{4}-\frac {1}{4}\right ) \operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-\frac {\left (\operatorname {csgn}\left (\sin \left (\frac {t}{2}\right )\right )-1\right ) \left (c_1 \operatorname {MathieuSPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+\operatorname {MathieuCPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )\right )}{4}\right )}{-2 c_1 \operatorname {MathieuSPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+2 c_1 \operatorname {MathieuS}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )+2 \operatorname {MathieuC}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )-2 \operatorname {MathieuCPrime}\left (-1, -2, \arccos \left (\cos \left (\frac {t}{2}\right )\right )\right )} \]

✗ 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