20.9.16 problem Problem 16

Internal problem ID [3760]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters Method. page 556
Problem number : Problem 16
Date solved : Monday, January 27, 2025 at 08:00:03 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+17 y&=\frac {64 \,{\mathrm e}^{-x}}{3+\sin \left (4 x \right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 70

dsolve(diff(y(x),x$2)+2*diff(y(x),x)+17*y(x)=64*exp(-x)/(3+sin(4*x)^2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.224 (sec). Leaf size: 72

DSolve[D[y[x],{x,2}]+2*D[y[x],x]+17*y[x]==64*Exp[-x]/(3+Sin[4*x]^2),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{3} e^{-x} \left (4 \sqrt {3} \sin (4 x) \arctan \left (\frac {\sin (4 x)}{\sqrt {3}}\right )+3 c_1 \sin (4 x)+3 \cos (4 x) (-\log (2-\cos (4 x))+\log (\cos (4 x)+2)+c_2)\right ) \]