74.7.4 problem 4

Internal problem ID [16010]
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 : 4
Date solved : Monday, March 31, 2025 at 02:26:53 PM
CAS classification : [[_homogeneous, `class D`], _Bernoulli]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 52

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

\begin{align*} y &= \frac {t}{\sqrt {2 t^{2} \cos \left (t \right )-4 \cos \left (t \right )-4 t \sin \left (t \right )+c_1}} \\ y &= -\frac {t}{\sqrt {2 t^{2} \cos \left (t \right )-4 \cos \left (t \right )-4 t \sin \left (t \right )+c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.248 (sec). Leaf size: 67

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

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

✓ Sympy. Time used: 1.949 (sec). Leaf size: 63

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

\[ \left [ y{\left (t \right )} = - t \sqrt {\frac {1}{C_{1} + 2 t^{2} \cos {\left (t \right )} - 4 t \sin {\left (t \right )} - 4 \cos {\left (t \right )}}}, \ y{\left (t \right )} = t \sqrt {\frac {1}{C_{1} + 2 t^{2} \cos {\left (t \right )} - 4 t \sin {\left (t \right )} - 4 \cos {\left (t \right )}}}\right ] \]