The theory of homogeneous linear equations is not limited to only second-order equations. It works for any order. No matter how many roots the characteristic polynomial has, the cases work the same way as for a 2nd order equation:- Any non-repeated root \( r \) will correspond to a term \( e^{rt} \) in the general solution.- A repeated root \( r \) with multiplicity \( n \) will correspond to \( n \) terms in the general solution: \( e^{rt} \), \( te^{rt} \), ..., \( t^{n-1}e^{rt} \).- A pair of complex conjugate roots \( a+bi \) and \( a-bi \) will correspond to the terms \( e^{at} \cos(bt) \) and \( e^{at} \sin(bt) \) in the general solution.3. Consider the differential equation \( y' + 2y = 0 \). a) The equation above is first-order linear homogeneous and therefore separable. Solve the equation by separation of variables. b) Because the equation is linear and homogeneous, it also has a characteristic polynomial. What is the characteristic polynomial? c) Solve the differential equation using its characteristic polynomial. Note that since the characteristic polynomial is linear, you should only have one root.

a) write y' =dydx The equation becomes dydx+2y=0 separate the variables and write it again by…

3. Consider the differential equation y' +2y= 0. a) The equation above is first order linear homogeneous and therefore separable. Solve the equation by separation of variables. b) Because the equation is linear and homogeneous, it also has a characteristic polynomial. What is the characteristic polynomial? c) Solve the differential equation using its characteristic polynomial. Note that since the characteristic polynomial is linear, you should only have one root.

3. Consider the differential equation y' +2y= 0. a) The equation above is first order linear homogeneous and therefore separable. Solve the equation by separation of variables. b) Because the equation is linear and homogeneous, it also has a characteristic polynomial. What is the characteristic polynomial? c) Solve the differential equation using its characteristic polynomial. Note that since the characteristic polynomial is linear, you should only have one root.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

(2.ry + r2)y = r² + 2y? solve the differential equation
Get a first-order differential equation to solve current in a LR circuit analysis. The voltage of 20sin(t) is applied. The inductance is0.8 henry resistor is 100 and the lo current is 0! As I journey into DQ I would like an in-depth solution. Show all work and rules, equations. I want to have the foundation.
Obtain the differential equation by eliminating all arbitrary constants. 1.) y² = Ax² + B secx
please ensure that it is typewritten
The deceleration of a certain tugboat in the water is directly proportional to the cube of the speed of the tug at any time t. The velocity of the tug is initially 5 m/s? and reduces to 3 m/s? after 2s. Show that the equation of motion of the tug is 225 V² = by using differential equations. 8t+9
Consider the differential equation: dy/dx = y(3 - y) 1) Use the slope field plotter to plot different solutions corresponding to A: (0,-2); B: (0,1); C: (0,3); D: (0,4). You can drag each point to the correct place if they're somewhere else on the plane, or click on the point and redefine. Capture a screenshot showing all the solution curves and insert below. f'(x)=y(3-y) Kin/5 -5 Density Length dy de -8 Step size 0.1 Ymas y (3-9) Input. ✔Solution A Solution B Solution C Solution D DCQ= AC: 2) To what value does y(x) approach, as x → ∞, for the solutions corresponding to B,C, and D? To what value does y(x) approach, as x →∞, for the solution corresponding to A? Why is it different from the others? Be specific and answer in terms of what you know about the slope field and its effect on the solution. - 3) Use wolframalpha.com or something similar to find the particular solution corresponding to each of the points A,B,C,D. How is the solution through Point C different from the other…
2) Solve the differential equation: (D² 11D + 28)y 48 e36x + e 4x =