72.16.33 problem 34

Internal problem ID [14850]
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 : 34
Date solved : Thursday, March 13, 2025 at 04:21:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=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: 0.016 (sec). Leaf size: 26

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

\[ y = \frac {7}{4}-\frac {3 t}{2}+\frac {t^{2}}{2}+\frac {{\mathrm e}^{-2 t}}{4}-2 \,{\mathrm e}^{-t} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 37

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

\[ y(t)\to \frac {1}{4} e^{-2 t} \left (e^{2 t} \left (2 t^2-6 t+7\right )-8 e^t+1\right ) \]

✓ Sympy. Time used: 0.220 (sec). Leaf size: 29

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

\[ y{\left (t \right )} = \frac {t^{2}}{2} - \frac {3 t}{2} + \frac {7}{4} - 2 e^{- t} + \frac {e^{- 2 t}}{4} \]