12.10.4 problem 4

Internal problem ID [1808]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 4
Date solved : Saturday, March 29, 2025 at 11:40:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 30

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

\[ y(x)\to e^x ((3 x+c_1) \sin (x)+\cos (x) (3 \log (\cos (x))+c_2)) \]

✓ Sympy. Time used: 0.361 (sec). Leaf size: 26

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

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