61.4.11 problem Problem 2(j)[k]

Internal problem ID [15336]
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 2(j)[k]
Date solved : Thursday, October 02, 2025 at 10:11:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-12 \\ y^{\prime }\left (0\right )&=7 \\ \end{align*}

✓ Maple. Time used: 0.093 (sec). Leaf size: 22

ode:=4*diff(diff(y(t),t),t)-4*diff(y(t),t)+y(t) = t^2; 
ic:=[y(0) = -12, D(y)(0) = 7]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = t^{2}+8 t +24+{\mathrm e}^{\frac {t}{2}} \left (17 t -36\right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 26

ode=4*D[y[t],{t,2}]-4*D[y[t],t]+y[t]==t^2; 
ic={y[0]==-12,Derivative[1][y][0] ==7}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to t^2+8 t+e^{t/2} (17 t-36)+24 \end{align*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2 + y(t) - 4*Derivative(y(t), t) + 4*Derivative(y(t), (t, 2)),0) 
ics = {y(0): -12, Subs(Derivative(y(t), t), t, 0): 7} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = t^{2} + 8 t + \left (17 t - 36\right ) e^{\frac {t}{2}} + 24 \]