14.4.4 problem 4

Internal problem ID [2522]
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 : 4
Date solved : Sunday, March 30, 2025 at 12:08:04 AM
CAS classification : [_Riccati]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.451 (sec). Leaf size: 16

ode:=diff(y(t),t) = y(t)^2+cos(t)^2; 
ic:=y(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {\operatorname {MathieuCPrime}\left (\frac {1}{2}, -\frac {1}{4}, t\right )}{\operatorname {MathieuC}\left (\frac {1}{2}, -\frac {1}{4}, t\right )} \]

✓ Mathematica. Time used: 0.214 (sec). Leaf size: 25

ode=D[y[t],t]==y[t]^2+Cos[t]^2; 
ic={y[0]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\frac {\text {MathieuCPrime}\left [\frac {1}{2},-\frac {1}{4},t\right ]}{\text {MathieuC}\left [\frac {1}{2},-\frac {1}{4},t\right ]} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t)**2 - cos(t)**2 + Derivative(y(t), t),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

NotImplementedError : The given ODE -y(t)**2 - cos(t)**2 + Derivative(y(t), t) cannot be solved by the lie group method