15.11.23 problem 23

Internal problem ID [3133]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 23
Date solved : Sunday, March 30, 2025 at 01:19:13 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+5 y&=2 x -{\mathrm e}^{-4 x}+\sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 43

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+5*y(x) = 2*x-exp(-4*x)+sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-2 x} \sin \left (x \right ) c_2 +{\mathrm e}^{-2 x} \cos \left (x \right ) c_1 +\frac {\sin \left (2 x \right )}{65}+\frac {2 x}{5}-\frac {8}{25}-\frac {8 \cos \left (2 x \right )}{65}-\frac {{\mathrm e}^{-4 x}}{5} \]

✓ Mathematica. Time used: 0.83 (sec). Leaf size: 59

ode=D[y[x],{x,2}]+4*D[y[x],x]+5*y[x]==2*x-Exp[-4*x]+Sin[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {2 x}{5}-\frac {e^{-4 x}}{5}+\frac {1}{65} \sin (2 x)-\frac {8}{65} \cos (2 x)+c_2 e^{-2 x} \cos (x)+c_1 e^{-2 x} \sin (x)-\frac {8}{25} \]

✓ Sympy. Time used: 0.451 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x + 5*y(x) - sin(2*x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + exp(-4*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {2 x}{5} + \left (C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )}\right ) e^{- 2 x} + \frac {\sin {\left (2 x \right )}}{65} - \frac {8 \cos {\left (2 x \right )}}{65} - \frac {8}{25} - \frac {e^{- 4 x}}{5} \]