Consider the differential equation y + a1 y + a2 y = 0, (8.2.13) where a1, a2 are

Chapter 8, Problem 43

(choose chapter or problem)

Consider the differential equation

\(y^{\prime \prime}+a_{1} y^{\prime}+a_{2} y=0\), (8.2.13)

where \(a_{1}, a_{2}\) are constants.

(a) If the auxiliary equation has real roots \(r_{1}\) and \(r_{2}\), what conditions on these roots would guarantee that every solution to Equation (8.2.13) satisfies

\(\lim _{x \rightarrow+\infty} y(x)=0 ?\)

(b) If the auxiliary equation has complex conjugate roots \(r=a \pm i b\), what conditions on these roots would guarantee that every solution to Equation (8.2.13) satisfies

\(\lim _{x \rightarrow+\infty} y(x)=0 ?\)

(c) If \(a_{1}, a_{2}\) are positive, prove that \(\lim _{x \rightarrow+\infty} y(x)=0\), for every solution to Equation (8.2.13).

(d) If \(a_{1}>0\) and \(a_{2}=0\), prove that all solutions to Equation (8.2.13) approach a constant value as \(x \rightarrow+\infty\).

(e) If \(a_{1}=0\) and \(a_{2}>0\), prove that all solutions to Equation (8.2.13) remain bounded as \(x \rightarrow+\infty\).