68.11.55 problem 68

Internal problem ID [17592]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 68
Date solved : Thursday, October 02, 2025 at 02:26:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 45

ode:=diff(diff(y(t),t),t)+diff(y(t),t)-2*y(t) = f(t); 
ic:=[y(0) = 0, D(y)(0) = a]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{-2 t} \left (\int _{0}^{t}f \left (\textit {\_z1} \right ) {\mathrm e}^{2 \textit {\_z1}}d \textit {\_z1} +a \right )}{3}+\frac {{\mathrm e}^{t} \left (\int _{0}^{t}{\mathrm e}^{-\textit {\_z1}} f \left (\textit {\_z1} \right )d \textit {\_z1} +a \right )}{3} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 123

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

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

✓ Sympy. Time used: 0.472 (sec). Leaf size: 61

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

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