90.14.16 problem 25

Internal problem ID [25244]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 3. Second Order Constant Coefficient Linear Differential Equations. Exercises at page 243
Problem number : 25
Date solved : Thursday, October 02, 2025 at 11:59:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }-6 y&={\mathrm e}^{3 t} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)-5*diff(y(t),t)-6*y(t) = exp(3*t); 
ic:=[y(0) = 2, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {10 \,{\mathrm e}^{6 t}}{21}+\frac {45 \,{\mathrm e}^{-t}}{28}-\frac {{\mathrm e}^{3 t}}{12} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 30

ode=D[y[t],{t,2}]-5*D[y[t],{t,1}]-6*y[t]==Exp[3*t]; 
ic={y[0]==2,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{84} e^{-t} \left (-7 e^{4 t}+40 e^{7 t}+135\right ) \end{align*}

✓ Sympy. Time used: 0.137 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-6*y(t) - exp(3*t) - 5*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {10 e^{6 t}}{21} - \frac {e^{3 t}}{12} + \frac {45 e^{- t}}{28} \]