Concept explainers
Program Description: Purpose of problem is to determine whether
Explanation of Solution
Given information:
The second order differential equation is,
The value of first order differential equation at
The value of the function at
Explanation:
Consider
The characteristic equation can be expressed as follows,
The roots of the equation are as follows,
Therefore, the general solution of the differential equation can be expressed as,
Apply the initial condition in equation
Therefore, the value of
Since,
Now,consider
Then, the given differential equation can be expressed as follows,
The characteristic equation can be expressed as follows,
The roots of the equation are as follows,
Therefore, the general solution of the differential equation can be expressed as,
Differentiate the above equation with respect to
Apply the initial condition
Apply the initial condition
Therefore, the value of
Then the value of
Since,
Now,consider
Then, the given differential equation can be expressed as follows,
The characteristic equation can be expressed as follows,
The roots of the equation are as follows,
Therefore, the general solution of the differential equation can be expressed as,
Differentiate the above equation with respect to
Apply the initial condition
Apply the initial condition
Since, the value of
For all odd value
Hence, the eigenvalues for the given equation are as follows,
Conclusion:
Thus, the Eigen functions corresponding to the eigenvalues are
Want to see more full solutions like this?
Chapter 3 Solutions
Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis
- (Conversion) An object’s polar moment of inertia, J, represents its resistance to twisting. For a cylinder, this moment of inertia is given by this formula: J=mr2/2+m( l 2 +3r 2 )/12misthecylindersmass( kg).listhecylinderslength(m).risthecylindersradius(m). Using this formula, determine the units for the cylinder’s polar moment of inertia.arrow_forwardSimplify this first line of the Boolean equation to just AB using simplifying theoremsarrow_forwardFind the value of φ(29), φ(32), φ(80), φ(100), φ(101), where φ() is the Euler’s Phi-Functionarrow_forward
- The quarter ring shown has a mass m and was cut from a thin, uniform plate. Knowing that ri = r2, determine the mass moment of inertia of the quarter ring with respect to (a) the axis AA', (b) the centroidal axis CC' that is perpendicular to the plane of the quarter ring. A' 12 B' C' Aarrow_forwardFind the derivative of the function. F(x) = -1/12/2 x2 f'(x) =arrow_forward2. calculates the trajectory r(t) and stores the coordinates for time steps At as a nested list trajectory that contains [[xe, ye, ze], [x1, y1, z1], [x2, y2, z2], ...]. Start from time t = 0 and use a time step At = 0.01; the last data point in the trajectory should be the time when the oscillator "hits the ground", i.e., when z(t) ≤ 0; 3. stores the time for hitting the ground (i.e., the first time t when z(t) ≤ 0) in the variable t_contact and the corresponding positions in the variables x_contact, y_contact, and z_contact. Print t_contact = 1.430 X_contact = 0.755 y contact = -0.380 z_contact = (Output floating point numbers with 3 decimals using format (), e.g., "t_contact = {:.3f}" .format(t_contact).) The partial example output above is for ze = 10. 4. calculates the average x- and y-coordinates 1 y = Yi N where the x, y, are the x(t), y(t) in the trajectory and N is the number of data points that you calculated. Store the result as a list in the variable center = [x_avg, y_avg]…arrow_forward
- An aluminum wire having a cross-sectional area equal to 4.60 x 10-6 m? carries a current of 7.50 A. The density of aluminum is 2.70 g/cm³. Assume each aluminum atom supplies one conduction electron per atom. Find the drift speed of the electrons in the wire. 1.95E-4 The equation for the drift velocity includes the number of charge carriers per volume, which in this case is equal to the number of atoms per volume. How do you calculate that if you know the density and the atomic weight of aluminum? mm/sarrow_forward1. Given a Boolean function E(02456) F(x,y, z) = а. Make a truth table b. Write an equation in the canonical SOP form. c. Write an equation in canonical POS form.arrow_forwardFor an object of mass m=3 kg to slide without friction up the rise of height h=1 m shown, it must have a minimum initial kinetic energy (in J) of: h O a. 40 O b. 20 O c. 30 O d. 10arrow_forward
- Find a transformation of a triangle A(1, 0) B(0, 1) C(1, 1) by one unit in x and y translating directions and then rotating 45' about the origin.arrow_forwardshow that the boolean equations is equivalent to the answerarrow_forwardPoint k at end of the rod as in Fig. slides along the fixed path (x-y/30), where x and y in (mm). y coordinate of k varies according to the relation y- 4i+5t (mm). take y=20 mm, determine (a) the velocity of k; and (b) the acceleration of k. 50 mmarrow_forward
- C++ for Engineers and ScientistsComputer ScienceISBN:9781133187844Author:Bronson, Gary J.Publisher:Course Technology Ptr