28.2.31 problem 31

Internal problem ID [4474]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 31
Date solved : Tuesday, March 04, 2025 at 06:46:10 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+5 y&=5 \,{\mathrm e}^{-x} \sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 48

ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)+5*y(x) = 5*exp(-x)*sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = {\mathrm e}^{-\frac {3 x}{2}} \cos \left (\frac {\sqrt {11}\, x}{2}\right ) c_{1} +{\mathrm e}^{-\frac {3 x}{2}} \sin \left (\frac {\sqrt {11}\, x}{2}\right ) c_{2} -2 \,{\mathrm e}^{-x} \left (\cos \left (2 x \right )+\frac {\sin \left (2 x \right )}{2}\right ) \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 62

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

\[ y(x)\to e^{-3 x/2} \left (-e^{x/2} (\sin (2 x)+2 \cos (2 x))+c_2 \cos \left (\frac {\sqrt {11} x}{2}\right )+c_1 \sin \left (\frac {\sqrt {11} x}{2}\right )\right ) \]

✓ Sympy. Time used: 0.353 (sec). Leaf size: 51

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

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