15.14.3 problem 3

Internal problem ID [3175]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 23, page 106
Problem number : 3
Date solved : Sunday, March 30, 2025 at 01:20:25 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 30

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 1/2*exp(x)+1/2*exp(-x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-2 x} \left (18 c_1 x +\sinh \left (3 x \right )+\cosh \left (3 x \right )+9 \,{\mathrm e}^{x}+18 c_2 \right )}{18} \]

✓ Mathematica. Time used: 0.287 (sec). Leaf size: 34

ode=D[y[x],{x,2}]+4*D[y[x],x]+4*y[x]==1/2*(Exp[x]+Exp[-x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{18} e^{-2 x} \left (9 e^x+e^{3 x}+18 (c_2 x+c_1)\right ) \]

✓ Sympy. Time used: 0.223 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - exp(x)/2 + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - exp(-x)/2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{- 2 x} - \frac {4 \sinh {\left (x \right )}}{9} + \frac {5 \cosh {\left (x \right )}}{9} \]