69.16.58 problem 531

Internal problem ID [18318]
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. Trial and error method. Exercises page 132
Problem number : 531
Date solved : Thursday, October 02, 2025 at 03:10:24 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 70

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

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

✓ Sympy. Time used: 0.217 (sec). Leaf size: 22

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

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