23.3.13 problem 8(c)

Internal problem ID [4154]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 4. The general linear differential equation of order n. Exercises at page 63
Problem number : 8(c)
Date solved : Sunday, March 30, 2025 at 02:41:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{2}+\frac {y}{8}&=\frac {\sin \left (x \right )}{8}-\frac {\cos \left (x \right )}{4} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)+1/2*diff(y(x),x)+1/8*y(x) = 1/8*sin(x)-1/4*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x}{4}} \sin \left (\frac {x}{4}\right ) c_2 +{\mathrm e}^{-\frac {x}{4}} \cos \left (\frac {x}{4}\right ) c_1 -\frac {3 \sin \left (x \right )}{13}+\frac {2 \cos \left (x \right )}{13} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 50

ode=D[y[x],{x,2}]+1/2*D[y[x],x]+1/8*y[x]==1/8*(Sin[x]-2*Cos[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {3 \sin (x)}{13}+\frac {2 \cos (x)}{13}+c_2 e^{-x/4} \cos \left (\frac {x}{4}\right )+c_1 e^{-x/4} \sin \left (\frac {x}{4}\right ) \]

✓ Sympy. Time used: 0.223 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)/8 - sin(x)/8 + cos(x)/4 + Derivative(y(x), x)/2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (\frac {x}{4} \right )} + C_{2} \cos {\left (\frac {x}{4} \right )}\right ) e^{- \frac {x}{4}} - \frac {3 \sin {\left (x \right )}}{13} + \frac {2 \cos {\left (x \right )}}{13} \]