89.26.3 problem 3

Internal problem ID [24870]
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 : 3
Date solved : Thursday, October 02, 2025 at 10:48:53 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.149 (sec). Leaf size: 30

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

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

✓ Sympy. Time used: 8.146 (sec). Leaf size: 56

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

\[ y{\left (x \right )} = \left (C_{1} - \log {\left (\cos {\left (e^{x} \right )} \right )}\right ) e^{x} + \left (C_{2} - \frac {\int \left (e^{x} \sec ^{2}{\left (e^{x} \right )} + 3 \tan {\left (e^{x} \right )}\right ) e^{3 x}\, dx}{2}\right ) e^{- x} + \frac {e^{2 x} \tan {\left (e^{x} \right )}}{2} \]