6.9.7 problem 7

Internal problem ID [1763]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 7
Date solved : Tuesday, September 30, 2025 at 05:19:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to e^x ((x+c_1) \sin (x)+\cos (x) (\log (\cos (x))+c_2)) \end{align*}

✓ Sympy. Time used: 0.287 (sec). Leaf size: 22

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

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