32.8.22 problem Exercise 21.29, page 231

Internal problem ID [5971]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 21. Undetermined Coefficients
Problem number : Exercise 21.29, page 231
Date solved : Sunday, March 30, 2025 at 10:28:35 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)-diff(y(x),x)-2*y(x) = 5*sin(x); 
ic:=y(0) = 1, D(y)(0) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-x}}{6}+\frac {{\mathrm e}^{2 x}}{3}+\frac {\cos \left (x \right )}{2}-\frac {3 \sin \left (x \right )}{2} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 30

ode=D[y[x],{x,2}]-D[y[x],x]-2*y[x]==5*Sin[x]; 
ic={y[0]==1,Derivative[1][y][0] ==-1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{6} \left (e^{-x}+2 e^{2 x}-9 \sin (x)+3 \cos (x)\right ) \]

✓ Sympy. Time used: 0.197 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x) - 5*sin(x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {e^{2 x}}{3} - \frac {3 \sin {\left (x \right )}}{2} + \frac {\cos {\left (x \right )}}{2} + \frac {e^{- x}}{6} \]