67.5.7 problem Problem 2(b)

Internal problem ID [14022]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6. Introduction to Systems of ODEs. Problems page 408
Problem number : Problem 2(b)
Date solved : Monday, March 31, 2025 at 08:22:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 21

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

\[ y = {\mathrm e}^{t} \left (\frac {1}{4}+c_2 \sin \left (2 t \right )+c_1 \cos \left (2 t \right )\right ) \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 57

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

\[ y(t)\to \frac {1}{2} e^t \left (2 \sin (2 t) \int _1^t\frac {1}{2} \cos (2 K[1])dK[1]+\cos (2 t) \left (\cos ^2(t)+2 c_2\right )+2 c_1 \sin (2 t)\right ) \]

✓ Sympy. Time used: 0.230 (sec). Leaf size: 22

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

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