69.18.20 problem 609

Internal problem ID [18395]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Initial value problem. Exercises page 140
Problem number : 609
Date solved : Thursday, October 02, 2025 at 03:11:20 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 29

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

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

✓ Mathematica. Time used: 0.229 (sec). Leaf size: 83

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

\begin{align*} y(x)&\to \frac {1}{2} e^{-x} \left (2 \sin (2 x) \int _1^x\frac {1}{2} e^{K[1]} \cos (2 K[1]) (4 \cos (2 K[1])+\sin (2 K[1]))dK[1]+2 c_1 \sin (2 x)+\cos (2 x) \left (-e^x \sin ^2(2 x)+2 c_2\right )\right ) \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (2 x \right )} + C_{2} \cos {\left (2 x \right )}\right ) e^{- x} + \sin {\left (2 x \right )} \]