62.3.2 problem 6.3 (b)

Internal problem ID [15436]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 6. Laplace transforms. Problems page 172
Problem number : 6.3 (b)
Date solved : Thursday, October 02, 2025 at 10:14:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+3 y&=f \left (t \right ) \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.121 (sec). Leaf size: 51

ode:=diff(diff(y(t),t),t)-4*diff(y(t),t)+3*y(t) = f(t); 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

\[ y = -\frac {{\mathrm e}^{3 t}}{2}+\frac {\int _{0}^{t}f \left (\textit {\_U1} \right ) {\mathrm e}^{3 t -3 \textit {\_U1}}d \textit {\_U1}}{2}-\frac {\int _{0}^{t}f \left (\textit {\_U1} \right ) {\mathrm e}^{t -\textit {\_U1}}d \textit {\_U1}}{2}+\frac {3 \,{\mathrm e}^{t}}{2} \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 117

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

\begin{align*} y(t)&\to -\frac {1}{2} e^t \left (-2 \int _1^t-\frac {1}{2} e^{-K[1]} f(K[1])dK[1]+2 e^{2 t} \int _1^0\frac {1}{2} e^{-3 K[2]} f(K[2])dK[2]-2 e^{2 t} \int _1^t\frac {1}{2} e^{-3 K[2]} f(K[2])dK[2]+2 \int _1^0-\frac {1}{2} e^{-K[1]} f(K[1])dK[1]+e^{2 t}-3\right ) \end{align*}

✓ Sympy. Time used: 0.635 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
y = Function("y") 
f = Function("f") 
ode = Eq(-f(t) + 3*y(t) - 4*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)
 

\[ y{\left (t \right )} = \left (\left (\frac {\int f{\left (t \right )} e^{- 3 t}\, dt}{2} - \frac {\int \limits ^{0} f{\left (t \right )} e^{- 3 t}\, dt}{2}\right ) e^{2 t} - \frac {\int f{\left (t \right )} e^{- t}\, dt}{2} + \frac {\int \limits ^{0} f{\left (t \right )} e^{- t}\, dt}{2}\right ) e^{t} \]