14.21.13 problem 13

Internal problem ID [2722]
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 : 13
Date solved : Tuesday, March 04, 2025 at 02:40:21 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }-4 y^{\prime }&=t +\cos \left (t \right )+2 \,{\mathrm e}^{-2 t} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

ode:=diff(diff(diff(y(t),t),t),t)-4*diff(y(t),t) = t+cos(t)+2*exp(-2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {\left (3+4 t -8 c_2 \right ) {\mathrm e}^{-2 t}}{16}-\frac {t^{2}}{8}+\frac {{\mathrm e}^{2 t} c_1}{2}+c_3 -\frac {\sin \left (t \right )}{5} \]

✓ Mathematica. Time used: 0.771 (sec). Leaf size: 51

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

\[ y(t)\to \frac {1}{16} e^{-2 t} \left (-2 e^{2 t} t^2+4 t+8 c_1 e^{4 t}+3-8 c_2\right )-\frac {\sin (t)}{5}+c_3 \]

✓ Sympy. Time used: 0.397 (sec). Leaf size: 31

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

\[ y{\left (t \right )} = C_{1} + C_{3} e^{2 t} - \frac {t^{2}}{8} + \left (C_{2} + \frac {t}{4}\right ) e^{- 2 t} - \frac {\sin {\left (t \right )}}{5} \]