91Ó°ÊÓ

The eigenvalues in Problems 1 through 5 are all nonnegative. First, determine whether \(\lambda=0\) is an eigenvalue; then find the positive eigenvalues and associated genuflections. y^{\prime \prime}+\lambda y=0 ; y^{\prime}(0)=0, y(1)=0

Express the solution of the given initial value problem as a sum of two oscillations as in Eq. (8). Throughout, primes denote derivatives with respect to time \(t\) , graph the solution function \(x(t)\) in such a way that you can identify and label its period. $$ x^{\prime \prime}+4 x=5 \sin 3 t ; x(0)=x^{\prime}(0)=0 $$

A homogeneous second-order linear differential equation, two functions \(y_{1}\) and \(y_{2}\), and a pair of initial conditions are given. First verify that \(y_{1}\) and \(y_{2}\) are solutions of the differential equation. Then find a particular solution of the form \(y=c_{1} y_{1}+c_{2} y_{2}\) that satisfies the given initial conditions. Primes denote derivatives with respect to \(x\). $$ y^{\prime \prime}-9 y=0 ; y_{1}=e^{3 x}, y_{2}=e^{-3 x} ; y(0)=-1, y^{\prime}(0)=15 $$

Prove that the eigenvalue problem $$ y^{\prime \prime}+\lambda y=0 ; \quad y(0)=0, \quad y(1)+y^{\prime}(1)=0 $$ has no negative eigenvalues. (Suggestion: Show graphically that the only root of the equation \(\tanh z=-z\) is \(z=0 .)\)

Suppose that the mass in a mass-spring-dashpot system with \(m=25, c=10\), and \(k=226\) is set in motion with \(x(0)=20\) and \(x^{\prime}(0)=41 .\) (a) Find the position function \(x(t)\) and show that its graph looks as indicated in Fig. \(3.4 .15 . \quad\) (b) Find the pseudoperiod of the oscillations and the equations of the "envelope curves" that are dashed in the figure.

The remaining problems in this section deal with free damped motion, a mass \(m\) is attached to both a spring (with given spring constant \(k\) ) and a dashpot (with given damping constant c). The mass is set in motion with initial position \(x_{0}\) and initial velocity \(v_{0}\). Find the position function \(x(t)\) and determine whether the motion is overdamped, critically damped, or underdamped. If it is underdamped, write the position function in the form \(x(t)=\) \(C_{1} e^{-p t} \cos \left(\omega_{1} t-\alpha_{1}\right) .\) Also, find the undamped position function \(u(t)=C_{0} \cos \left(\omega_{0} t-\alpha_{0}\right)\) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so \(c=0) .\) Finally, construct a figure that illustrates the effect of damping by comparing the graphs of \(x(t)\) and \(u(t)\). \(m=\frac{1}{2}, c=3, k=4 ; x_{0}=2, v_{0}=0\)

In Problems 13 through 20, a third-order homogeneous linear equation and three linearly independent solutions are given. Find a particular solution satisfying the given initial conditions. $$ \begin{aligned} &y^{(3)}+9 y^{\prime}=0 ; y(0)=3, y^{\prime}(0)=-1, y^{\prime \prime}(0)=2 ; y_{1}=1, \\ &v_{2}=\cos 3 x, y_{3}=\sin 3 x \end{aligned} $$

Solve the initial value problems given in Problems. $$ 3 y^{(3)}+2 y^{\prime \prime}=0 ; y(0)=-1, y^{\prime}(0)=0, y^{\prime \prime}(0)=1 $$

Problems pertain to the solution of differential equations with complex coefficients. Use the quadratic formula to solve the following equations. Note in each case that the roots are not complex conjugates. (a) \(x^{2}+i x+2=0\) (b) \(x^{2}-2 i x+3=0\)

Make the substitution \(v=\ln x\) of Problem 51 to find general solutions (for \(x>0\) ) of the Euler equations in Problems. $$ x^{2} y^{\prime \prime}+x y^{\prime}+9 y=0 $$