69.17.13 problem 563

Internal problem ID [18349]
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. Superposition principle. Exercises page 137
Problem number : 563
Date solved : Thursday, October 02, 2025 at 03:10:44 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 5 y+2 y^{\prime }+y^{\prime \prime }&=4 \,{\mathrm e}^{-x}+17 \sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+5*y(x) = 4*exp(-x)+17*sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\left (c_1 +1\right ) \cos \left (2 x \right )+\sin \left (2 x \right ) c_2 +1\right ) {\mathrm e}^{-x}-4 \cos \left (2 x \right )+\sin \left (2 x \right ) \]

✓ Mathematica. Time used: 0.214 (sec). Leaf size: 94

ode=D[y[x],{x,2}]+2*D[y[x],x]+5*y[x]==4*Exp[-x]+17*Sin[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x} \left (\cos (2 x) \int _1^x-\frac {1}{2} \sin (2 K[2]) \left (17 e^{K[2]} \sin (2 K[2])+4\right )dK[2]+\sin (2 x) \int _1^x\frac {1}{2} \cos (2 K[1]) \left (17 e^{K[1]} \sin (2 K[1])+4\right )dK[1]+c_2 \cos (2 x)+c_1 \sin (2 x)\right ) \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*y(x) - 17*sin(2*x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 4*exp(-x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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