74.7.3 problem 3

Internal problem ID [16009]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 3
Date solved : Monday, March 31, 2025 at 02:26:04 PM
CAS classification : [_Bernoulli]

\begin{align*} 2 t y^{\prime }-y&=2 t y^{3} \cos \left (t \right ) \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 70

ode:=2*t*diff(y(t),t)-y(t) = 2*t*y(t)^3*cos(t); 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= \frac {\sqrt {-2 t \left (t \sin \left (t \right )-\frac {c_1}{2}+\cos \left (t \right )\right )}}{2 t \sin \left (t \right )+2 \cos \left (t \right )-c_1} \\ y &= -\frac {\sqrt {-2 t \left (t \sin \left (t \right )-\frac {c_1}{2}+\cos \left (t \right )\right )}}{2 t \sin \left (t \right )+2 \cos \left (t \right )-c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.28 (sec). Leaf size: 71

ode=2*t*D[y[t],t]-y[t]==2*t*y[t]^3*Cos[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to -\frac {\sqrt {t}}{\sqrt {-2 \int _1^t\cos (K[1]) K[1]dK[1]+c_1}} \\ y(t)\to \frac {\sqrt {t}}{\sqrt {-2 \int _1^t\cos (K[1]) K[1]dK[1]+c_1}} \\ y(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 1.503 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t*y(t)**3*cos(t) + 2*t*Derivative(y(t), t) - y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \left [ y{\left (t \right )} = - \sqrt {- \frac {t}{C_{1} + 2 t \sin {\left (t \right )} + 2 \cos {\left (t \right )}}}, \ y{\left (t \right )} = \sqrt {- \frac {t}{C_{1} + 2 t \sin {\left (t \right )} + 2 \cos {\left (t \right )}}}\right ] \]