61.5.11 problem Problem 2(f)

Internal problem ID [15378]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number : Problem 2(f)
Date solved : Thursday, October 02, 2025 at 10:12:22 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 49

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

\[ y = \frac {{\mathrm e}^{t} \left (-c_2 \sqrt {3}+c_1 \right ) \cos \left (\sqrt {3}\, t \right )}{4}+\frac {{\mathrm e}^{t} \left (c_1 \sqrt {3}+c_2 \right ) \sin \left (\sqrt {3}\, t \right )}{4}+c_3 -\frac {3 \cos \left (t \right )}{13}+\frac {2 \sin \left (t \right )}{13} \]

✓ Mathematica. Time used: 60.311 (sec). Leaf size: 130

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

\begin{align*} y(t)&\to \int _1^te^{K[3]} \left (c_2 \cos \left (\sqrt {3} K[3]\right )+\int _1^{K[3]}-\frac {e^{-K[2]} \sin (K[2]) \sin \left (\sqrt {3} K[2]\right )}{\sqrt {3}}dK[2] \cos \left (\sqrt {3} K[3]\right )+c_1 \sin \left (\sqrt {3} K[3]\right )+\sin \left (\sqrt {3} K[3]\right ) \int _1^{K[3]}\frac {e^{-K[1]} \cos \left (\sqrt {3} K[1]\right ) \sin (K[1])}{\sqrt {3}}dK[1]\right )dK[3]+c_3 \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 41

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

\[ y{\left (t \right )} = C_{1} + \left (C_{2} \sin {\left (\sqrt {3} t \right )} + C_{3} \cos {\left (\sqrt {3} t \right )}\right ) e^{t} + \frac {2 \sin {\left (t \right )}}{13} - \frac {3 \cos {\left (t \right )}}{13} \]