89.19.28 problem 28

Internal problem ID [24756]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 151
Problem number : 28
Date solved : Thursday, October 02, 2025 at 10:47:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+7 y^{\prime }+12 y&={\mathrm e}^{-3 x} \sec \left (x \right )^{2} \left (1+2 \tan \left (x \right )\right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+7*diff(y(x),x)+12*y(x) = exp(-3*x)*sec(x)^2*(1+2*tan(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_2 +\tan \left (x \right )-1\right ) {\mathrm e}^{-3 x}+{\mathrm e}^{-4 x} c_1 \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 26

ode=D[y[x],{x,2}]+7*D[y[x],{x,1}]+12*y[x]== Exp[-3*x]*Sec[x]^2*( 1+2*Tan[x] ); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-4 x} \left (e^x \tan (x)+c_2 e^x+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.818 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(2*tan(x) + 1)*exp(-3*x)*sec(x)**2 + 12*y(x) + 7*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} e^{- x} - \frac {\sqrt {2} \cos {\left (2 x + \frac {\pi }{4} \right )}}{\cos {\left (2 x \right )} + 1} - \frac {1}{\cos {\left (2 x \right )} + 1}\right ) e^{- 3 x} \]