65.9.2 problem 16.1 (ii)

Internal problem ID [13702]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 16, Higher order linear equations with constant coefficients. Exercises page 153
Problem number : 16.1 (ii)
Date solved : Wednesday, March 05, 2025 at 10:13:18 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-3 y^{\prime \prime }+2 y&=\sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 38

ode:=diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+2*y(x) = sin(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} c_{1} +c_{2} {\mathrm e}^{\left (1+\sqrt {3}\right ) x}+c_{3} {\mathrm e}^{-\left (\sqrt {3}-1\right ) x}+\frac {\cos \left (x \right )}{26}+\frac {5 \sin \left (x \right )}{26} \]

✓ Mathematica. Time used: 0.101 (sec). Leaf size: 143

ode=D[y[x],{x,3}]-3*D[y[x],{x,2}]+2*y[x]==Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{x-\sqrt {3} x} \int _1^x\frac {1}{6} e^{\left (-1+\sqrt {3}\right ) K[1]} \sin (K[1])dK[1]+e^{\left (1+\sqrt {3}\right ) x} \int _1^x\frac {1}{6} e^{-\left (\left (1+\sqrt {3}\right ) K[2]\right )} \sin (K[2])dK[2]+e^x \int _1^x-\frac {1}{3} e^{-K[3]} \sin (K[3])dK[3]+c_1 e^{x-\sqrt {3} x}+c_2 e^{\left (1+\sqrt {3}\right ) x}+c_3 e^x \]

✓ Sympy. Time used: 0.138 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*y(x) - sin(x) - 3*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + C_{2} e^{- \sqrt {3} x} + C_{3} e^{\sqrt {3} x}\right ) e^{x} + \frac {5 \sin {\left (x \right )}}{26} + \frac {\cos {\left (x \right )}}{26} \]