14.4.15 problem 15

Internal problem ID [2533]
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 : 15
Date solved : Sunday, March 30, 2025 at 12:09:02 AM
CAS classification : [_Bernoulli]

\begin{align*} y^{\prime }&=\frac {\left (1+\cos \left (4 t \right )\right ) y}{4}-\frac {\left (1-\cos \left (4 t \right )\right ) y^{2}}{800} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=100 \end{align*}

✓ Maple. Time used: 0.815 (sec). Leaf size: 45

ode:=diff(y(t),t) = 1/4*(1+cos(4*t))*y(t)-1/800*(1-cos(4*t))*y(t)^2; 
ic:=y(0) = 100; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {800 \,{\mathrm e}^{\frac {t}{4}+\frac {\sin \left (4 t \right )}{16}}}{-8+\int _{0}^{t}{\mathrm e}^{\frac {\textit {\_z1}}{4}+\frac {\sin \left (4 \textit {\_z1} \right )}{16}} \left (-1+\cos \left (4 \textit {\_z1} \right )\right )d \textit {\_z1}} \]

✓ Mathematica. Time used: 0.745 (sec). Leaf size: 61

ode=D[y[t],t]==1/4*(1+Cos[4*t])*y[t]-1/800*(1-Cos[4*t])*y[t]^2; 
ic={y[0]==100}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to -\frac {100 e^{\frac {1}{16} (4 t+\sin (4 t))}}{100 \int _0^t-\frac {1}{400} e^{\frac {1}{16} (4 K[1]+\sin (4 K[1]))} \sin ^2(2 K[1])dK[1]-1} \]

✓ Sympy. Time used: 31.955 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq((1 - cos(4*t))*y(t)**2/800 - (cos(4*t) + 1)*y(t)/4 + Derivative(y(t), t),0) 
ics = {y(0): 100} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {800 e^{\frac {t}{4} + \frac {\sin {\left (4 t \right )}}{16}}}{\int e^{\frac {t}{4}} e^{\frac {\sin {\left (4 t \right )}}{16}}\, dt - \int \limits ^{0} e^{\frac {t}{4}} e^{\frac {\sin {\left (4 t \right )}}{16}}\, dt - \int e^{\frac {t}{4}} e^{\frac {\sin {\left (4 t \right )}}{16}} \cos {\left (4 t \right )}\, dt + \int \limits ^{0} e^{\frac {t}{4}} e^{\frac {\sin {\left (4 t \right )}}{16}} \cos {\left (4 t \right )}\, dt + 8} \]