67.15.35 problem 22.10 (h)

Internal problem ID [16753]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 22. Method of undetermined coefficients. Additional exercises page 412
Problem number : 22.10 (h)
Date solved : Thursday, October 02, 2025 at 01:38:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+5 y&={\mathrm e}^{-x} \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 34

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

\[ y = \frac {{\mathrm e}^{-x} \left (3 \sin \left (x \right )+2 \cos \left (x \right )\right )}{39}+{\mathrm e}^{2 x} \left (\cos \left (x \right ) c_1 +\sin \left (x \right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 68

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

\begin{align*} y(x)&\to e^{2 x} \left (\cos (x) \int _1^x-e^{-3 K[2]} \sin ^2(K[2])dK[2]+\sin (x) \int _1^xe^{-3 K[1]} \cos (K[1]) \sin (K[1])dK[1]+c_2 \cos (x)+c_1 \sin (x)\right ) \end{align*}

✓ Sympy. Time used: 0.200 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )}\right ) e^{2 x} + \frac {e^{- x} \sin {\left (x \right )}}{13} + \frac {2 e^{- x} \cos {\left (x \right )}}{39} \]