89.26.6 problem 7

Internal problem ID [24873]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 11. Variation of parameters and other methods. Miscellaneous Exercises at page 177
Problem number : 7
Date solved : Thursday, October 02, 2025 at 10:48:56 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 142

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

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

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 30

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

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

✓ Sympy. Time used: 1.082 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = \left (C_{1} + \left (C_{2} + \int e^{- 2 x} \sec ^{2}{\left (e^{- x} \right )}\, dx\right ) e^{x} + \tan {\left (e^{- x} \right )}\right ) e^{x} \]