75.18.19 problem 608

Internal problem ID [17036]
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 : 608
Date solved : Monday, March 31, 2025 at 03:38:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

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

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

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

✓ Mathematica. Time used: 0.088 (sec). Leaf size: 68

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

\[ y(x)\to e^{2 x} \left (\cos (x) \int _1^x-e^{-2 K[2]} \sin ^2(K[2])dK[2]+\sin (x) \int _1^xe^{-2 K[1]} \cos (K[1]) \sin (K[1])dK[1]+c_2 \cos (x)+c_1 \sin (x)\right ) \]

✓ Sympy. Time used: 0.225 (sec). Leaf size: 27

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

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