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 : Monday, January 27, 2025 at 08:40:07 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*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 33

dsolve(diff(y(x),x$2)+1/2*diff(y(x),x)+1/8*y(x)=1/8*(sin(x)-2*cos(x)),y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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