86.7.6 problem 6

Internal problem ID [23153]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5c at page 83
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:23:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+5 y&=4 \,{\mathrm e}^{3 t} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

ode:=diff(diff(y(t),t),t)+3*diff(y(t),t)+5*y(t) = 4*exp(3*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {3 t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right ) c_2 +{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) c_1 +\frac {4 \,{\mathrm e}^{3 t}}{23} \]

✓ Mathematica. Time used: 0.017 (sec). Leaf size: 56

ode=D[y[t],{t,2}]+3*D[y[t],t]+5*y[t]==4*Exp[3*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {1}{23} e^{-3 t/2} \left (4 e^{9 t/2}+23 c_2 \cos \left (\frac {\sqrt {11} t}{2}\right )+23 c_1 \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.137 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(5*y(t) - 4*exp(3*t) + 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {11} t}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {11} t}{2} \right )}\right ) e^{- \frac {3 t}{2}} + \frac {4 e^{3 t}}{23} \]