20.9.6 problem Problem 6

Internal problem ID [3750]
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 6
Date solved : Tuesday, March 04, 2025 at 05:10:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+5 y&={\mathrm e}^{2 x} \tan \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+5*y(x) = exp(2*x)*tan(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 29

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

\[ y(x)\to e^{2 x} (\cos (x) (-\text {arctanh}(\sin (x)))+c_2 \cos (x)+c_1 \sin (x)) \]

✓ Sympy. Time used: 0.497 (sec). Leaf size: 34

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

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