20.9.15 problem Problem 15

Internal problem ID [3759]
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 15
Date solved : Monday, January 27, 2025 at 08:00:01 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+y&=\frac {{\mathrm e}^{-x}}{\sqrt {-x^{2}+4}} \end{align*}

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 49

dsolve(diff(y(x),x$2)+2*diff(y(x),x)+y(x)=exp(-x)/sqrt(4-x^2),y(x), singsol=all)
 

\[ y \left (x \right ) = -\frac {\left (\left (-c_{1} x -\arcsin \left (\frac {x}{2}\right ) x -c_{2} \right ) \sqrt {-x^{2}+4}+x^{2}-4\right ) {\mathrm e}^{-x}}{\sqrt {-x^{2}+4}} \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 45

DSolve[D[y[x],{x,2}]+2*D[y[x],x]+y[x]==Exp[-x]/Sqrt[4-x^2],y[x],x,IncludeSingularSolutions -> True]
 

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