28.2.13 problem 13

Internal problem ID [4456]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 13
Date solved : Sunday, March 30, 2025 at 03:23:02 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y^{\prime }+10 y&=3 x \,{\mathrm e}^{-3 x}-2 \,{\mathrm e}^{3 x} \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)+6*diff(y(x),x)+10*y(x) = 3*x*exp(-3*x)-2*exp(3*x)*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-3 x} \left (c_2 \sin \left (x \right )+c_1 \cos \left (x \right )+\frac {\left (-3 \cos \left (x \right )-\sin \left (x \right )\right ) {\mathrm e}^{6 x}}{60}+3 x \right ) \]

✓ Mathematica. Time used: 0.397 (sec). Leaf size: 46

ode=D[y[x],{x,2}]+6*D[y[x],x]+10*y[x]==3*x*Exp[-3*x]-2*Exp[3*x]*Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{60} e^{-3 x} \left (180 x-3 \left (e^{6 x}-20 c_2\right ) \cos (x)-\left (e^{6 x}-60 c_1\right ) \sin (x)\right ) \]

✓ Sympy. Time used: 0.497 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*exp(-3*x) + 10*y(x) + 2*exp(3*x)*cos(x) + 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\left (- \sin {\left (x \right )} - 3 \cos {\left (x \right )}\right ) e^{3 x}}{60} + \left (C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )} + 3 x\right ) e^{- 3 x} \]