76.15.7 problem 7

Internal problem ID [17569]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.5 (Nonhomogeneous Equations, Method of Undetermined Coefficients). Problems at page 260
Problem number : 7
Date solved : Thursday, March 13, 2025 at 10:13:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+4 y&=2 \,{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 20

ode:=diff(diff(y(t),t),t)-5*diff(y(t),t)+4*y(t) = 2*exp(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{4 t} c_{1} -\frac {2 \,{\mathrm e}^{t} \left (t -\frac {3 c_{2}}{2}\right )}{3} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 30

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

\[ y(t)\to \frac {1}{9} e^t \left (-6 t+9 c_2 e^{3 t}-2+9 c_1\right ) \]

✓ Sympy. Time used: 0.193 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = \left (C_{1} + C_{2} e^{3 t} - \frac {2 t}{3}\right ) e^{t} \]