61.4.38 problem Problem 13(b)

Internal problem ID [15363]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 13(b)
Date solved : Thursday, October 02, 2025 at 10:12:08 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }-y^{\prime }+2 y&=4 t \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=-2 \\ y^{\prime \prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.099 (sec). Leaf size: 17

ode:=diff(diff(diff(y(t),t),t),t)-2*diff(diff(y(t),t),t)-diff(y(t),t)+2*y(t) = 4*t; 
ic:=[y(0) = 2, D(y)(0) = -2, (D@@2)(y)(0) = 4]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -6 \sinh \left (t \right )+1+2 t +{\mathrm e}^{2 t} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 27

ode=D[ y[t],{t,3}]-2*D[y[t],{t,2}]-D[y[t],t]+2*y[t]==4*t; 
ic={y[0]==2,Derivative[1][y][0] ==-2,Derivative[2][y][0] ==4}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to 2 t+3 e^{-t}-3 e^t+e^{2 t}+1 \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 22

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

\[ y{\left (t \right )} = 2 t + e^{2 t} - 3 e^{t} + 1 + 3 e^{- t} \]