91Ó°ÊÓ

Prove that if \(\phi_{1}\) and \(\phi_{2}\) are solutions of \(L[y]=g_{1}(x)\) and \(L[y]=g_{2}(x)\), respectively, on an interval \(I\), then \(\phi_{1}+\phi_{2}\) is a solution of \(L[y]=g_{1}+\) \(g_{2}\).

The free damped motion of a mass on a spring at time \(t\) is governed by the equation $$m \ddot{y}+c \dot{y}+k y=0,$$ where the coefficients are constants. The dot, as usual, denotes differentiation with respect to time. The roots of the characteristic equation are $$\lambda_{1,2}=\frac{-c \pm \sqrt{c^{2}-4 m k}}{2 m}$$ Describe the behavior of the solution in the three different cases of \(c^{2}-4 m k\) positive, negative or zero.

Show that every nontrivial solution of the equation $$y^{\prime \prime}+(\sinh x) y=0$$ has at most one zero in \((-\infty, 0)\) and infinitely many zeros in \((0, \infty)\).

Suppose that \(q(x)>0\) and \(q(x)\) is continuous in the interval \((0, \infty)\). Prove that every nontrivial solution of $$y^{\prime \prime}+q(x) y=0$$ has infinitely many zeros in \((0, \infty)\).

(Sonin-Polya theorem). Let \(p(x)>0\) and \(q(x) \neq 0\) be continuously differentiable on an interval \(I\). If \(p(x) q(x)\) is nonincreasing (nondecreasing) on \(I\), then the absolute values of the relative maxima and minima of every nontrivial solution of the equation $$\frac{d}{d x}\left(p(x) \frac{d y}{d x}\right)+q(x) y=0$$ are nondecreasing (nonincreasing) as \(x\) increases.