84.37.10 problem 26.10

Internal problem ID [22352]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 26. Solutions of linear differential equations with constant coefficients by Laplace transform. Solved problems. Page 159
Problem number : 26.10
Date solved : Thursday, October 02, 2025 at 08:37:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&=f \left (t \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.150 (sec). Leaf size: 39

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t)+2*y(t) = f(t); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 90

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

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

✓ Sympy. Time used: 0.586 (sec). Leaf size: 46

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

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