14.21.12 problem 12

Internal problem ID [2721]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.15, Higher order equations. Excercises page 263
Problem number : 12
Date solved : Tuesday, March 04, 2025 at 02:40:20 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+y^{\prime }&=2 t^{2}+4 \sin \left (t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

ode:=diff(diff(diff(y(t),t),t),t)+diff(y(t),t) = 2*t^2+4*sin(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (-c_2 -2\right ) \cos \left (t \right )+\left (c_1 -2 t \right ) \sin \left (t \right )+\frac {2 t^{3}}{3}-4 t +c_3 \]

✓ Mathematica. Time used: 0.219 (sec). Leaf size: 35

ode=D[y[t],{t,3}]+D[y[t],t]==2*t^2+4*Sin[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {2 t^3}{3}-4 t-(2+c_2) \cos (t)+(-2 t+c_1) \sin (t)+c_3 \]

✓ Sympy. Time used: 0.191 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-2*t**2 - 4*sin(t) + Derivative(y(t), t) + Derivative(y(t), (t, 3)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} + C_{3} \cos {\left (t \right )} + \frac {2 t^{3}}{3} - 4 t + \left (C_{2} - 2 t\right ) \sin {\left (t \right )} \]