81.11.16 problem 15-15

Internal problem ID [21640]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 15. Method of undetermined coefficients. Page 337.
Problem number : 15-15
Date solved : Thursday, October 02, 2025 at 07:59:20 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 5 y+2 y^{\prime }+y^{\prime \prime }&=3 \sin \left (x +\frac {\pi }{4}\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 35

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+5*y(x) = 3*sin(x+1/4*Pi); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\cos \left (2 x \right ) c_1 +c_2 \sin \left (2 x \right )\right ) {\mathrm e}^{-x}+\frac {3 \sqrt {2}\, \left (\cos \left (x \right )+3 \sin \left (x \right )\right )}{20} \]

✓ Mathematica. Time used: 0.216 (sec). Leaf size: 55

ode=D[y[x],{x,2}]+2*D[y[x],x]+5*y[x]==3*Cos[x-Pi/4]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{20} e^{-x} \left (9 \sqrt {2} e^x \sin (x)+20 c_2 \cos (2 x)+\cos (x) \left (3 \sqrt {2} e^x+40 c_1 \sin (x)\right )\right ) \end{align*}

✓ Sympy. Time used: 0.175 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x) - 3*sin(x + pi/4) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (2 x \right )} + C_{2} \cos {\left (2 x \right )}\right ) e^{- x} + \frac {3 \sin {\left (x + \frac {\pi }{4} \right )}}{5} - \frac {3 \cos {\left (x + \frac {\pi }{4} \right )}}{10} \]