72.16.32 problem 33

Internal problem ID [14849]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 4. Forcing and Resonance. Section 4.1 page 399
Problem number : 33
Date solved : Thursday, March 13, 2025 at 04:21:44 AM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }&=3 t +2 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)+4*diff(y(t),t) = 3*t+2; 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {3 t^{2}}{8}+\frac {5 \,{\mathrm e}^{-4 t}}{64}+\frac {5 t}{16}-\frac {5}{64} \]

✓ Mathematica. Time used: 4.52 (sec). Leaf size: 124

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

\[ y(t)\to \int _1^te^{-4 K[2]} \left (\int _1^{K[2]}e^{4 K[1]} (3 K[1]+2)dK[1]-\int _1^0e^{4 K[1]} (3 K[1]+2)dK[1]\right )dK[2]-\int _1^0e^{-4 K[2]} \left (\int _1^{K[2]}e^{4 K[1]} (3 K[1]+2)dK[1]-\int _1^0e^{4 K[1]} (3 K[1]+2)dK[1]\right )dK[2] \]

✓ Sympy. Time used: 0.216 (sec). Leaf size: 26

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

\[ y{\left (t \right )} = \frac {3 t^{2}}{8} + \frac {5 t}{16} - \frac {5}{64} + \frac {5 e^{- 4 t}}{64} \]