20.9.9 problem Problem 9

Internal problem ID [3753]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters Method. page 556
Problem number : Problem 9
Date solved : Sunday, March 30, 2025 at 02:07:23 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+13 y&=4 \,{\mathrm e}^{3 x} \sec \left (2 x \right )^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 38

ode:=diff(diff(y(x),x),x)-6*diff(y(x),x)+13*y(x) = 4*exp(3*x)*sec(2*x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{3 x} \left (c_2 \sin \left (2 x \right )+c_1 \cos \left (2 x \right )+\ln \left (\sec \left (2 x \right )+\tan \left (2 x \right )\right ) \sin \left (2 x \right )-1\right ) \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 37

ode=D[y[x],{x,2}]-6*D[y[x],x]+13*y[x]==4*Exp[3*x]*Sec[2*x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{3 x} \left (c_2 \cos (2 x)+\sin (2 x) \coth ^{-1}(\sin (2 x))+c_1 \sin (2 x)-1\right ) \]

✓ Sympy. Time used: 0.725 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(13*y(x) - 4*exp(3*x)/cos(2*x)**2 - 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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