13.5.12 problem 15

Internal problem ID [2358]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.10. Page 80
Problem number : 15
Date solved : Monday, January 27, 2025 at 05:49:58 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*}

✓ Solution by Maple

Time used: 1.031 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

Time used: 14.322 (sec). Leaf size: 61

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

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