Internal problem ID [2788]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=\frac {{\mathrm e}^{-x}}{\sqrt {-x^{2}+4}}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 54
dsolve(diff(y(x),x$2)+2*diff(y(x),x)+y(x)=exp(-x)/sqrt(4-x^2),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-x} c_{2} +x \,{\mathrm e}^{-x} c_{1} -\frac {{\mathrm e}^{-x} \left (-\arcsin \left (\frac {x}{2}\right ) x \sqrt {-x^{2}+4}+x^{2}-4\right )}{\sqrt {-x^{2}+4}} \]
✓ Solution by Mathematica
Time used: 0.085 (sec). Leaf size: 50
DSolve[y''[x]+2*y'[x]+y[x]==Exp[-x]/Sqrt[4-x^2],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} \left (-2 x \arctan \left (\frac {\sqrt {4-x^2}}{x+2}\right )+\sqrt {4-x^2}+c_2 x+c_1\right ) \]