Consider ỹ + cỷ +y = 0, and assume y(0) = 1, ÿ(0) = 0. (a)...

Consider ỹ + cỷ +y = 0, and assume y(0) = 1, ÿ(0) = 0. (a) Why is this a homogeneous system?
(b) What would change if this was to be an inhomogeneous system? When is this inhomogeneous aspect applied - this is a technical point, but helps us to understand that the initial condition is an external input too, but is *only* applied at t = 0?
(c) Choose e to make this problem overdamped/critically damped/underdamped.
(d) Follow the 3-step procedure to solve this homogeneous system for the ho- Imogeneous solution y which we call y: Step 1: Get yh. Step 2: Satisfy condition, and Step 3: Sketch. • Solve the overdamped case where c is the critically damped version where c is increased by 1 past the critically damped value. Solve the critically damped case, • Solve the underdamped case where e is the critically damped version where c is reduced by 1 below the critically damped value.

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Colby Messinger answered 1 year ago

Consider the IVPs: (A) y'+2y = 1, 0<t<1 , y(0)=2. (B) y' = y(1-y), 0<t<1 ,...

Consider the IVPs: (A) y'+2y = 1, 0<t<1 , y(0)=2. (B) y' = y(1-y), 0<t<1 , y(0)=1/2. 1. For each one, do the following: a. Find the exact solution y(t) and evaluate it at t=1. b. Apply Euler's method with Δt=1/4 to find Y4 ≈ y(1). Make a table of tn, Yn for n=0,1,2,3,4. c. Find the error at t=1. 2. Euler's method is obtained by approximating y'(tn) by a forward finite difference. Use the backward difference approximation to y'(tn+1)...

Consider the following second-order ODE, y"+1/4y=0 with y(0) = 1 and y'(0)=0. Transform this unique equation...

Consider the following second-order ODE, y"+1/4y=0 with y(0) = 1 and y'(0)=0. Transform this unique equation into a system of two 1st-order ODEs. Solve the obtained system for t in [0,0.6] with h = 0.2 by MATLAB using Euler Method or Improved Euler Method.

y''-2y'+1=0; y(0)=0, y'(0)=-1

la place transform of 1. y´´+4y´+3y=0. y(0)=3, y´(0)=1 2. y´´+2y´+y=0. y(0)=1, y´(0)=1

Consider the following initial value problem. y''−4y = 0, y(0) = 0, y'(0) = 5 (a)...

Consider the following initial value problem. y''−4y = 0, y(0) = 0, y'(0) = 5 (a) Solve the IVP using the characteristic equation method from chapter 4. (b) Solve the IVP using the Laplace transform method from chapter 7. (Hint: If you don’t have the same ﬁnal answer for each part, you’ve done something wrong.)

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Consider ỹ + cỷ +y = 0, and assume y(0) = 1, ÿ(0) = 0. (a)...

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Consider the IVPs: (A) y'+2y = 1, 0<t<1 , y(0)=2. (B) y' = y(1-y), 0<t<1 ,...

Consider the following second-order ODE, y"+1/4y=0 with y(0) = 1 and y'(0)=0. Transform this unique equation...

y''-2y'+1=0; y(0)=0, y'(0)=-1

la place transform of 1. y´´+4y´+3y=0. y(0)=3, y´(0)=1 2. y´´+2y´+y=0. y(0)=1, y´(0)=1

Consider the following initial value problem. y''−4y = 0, y(0) = 0, y'(0) = 5 (a)...

y"´-9y"+26y´-24y=1 y(0)=y´(0)=y"(0)=1

y''' + 2y'' − y' − 2y = sin(4t), y(0) = 0, y'(0) = 0, y''(0) = 1

Solve y'''-4y'=32xe2x-24x2 , y(0)=0 y'(0)=0 y''(0)=-1

Solve the following IVPs: A. y"" +2y′′+y = 4et, y(0)=1, y′(0)=1, y′′(0)=1, y′′′(0)=1. B. y′′′+25y′ = 325e−t,...

la place transform of 1. y´´+4y´+6y_=0, y(0)=1, y´(0)=-4 2. y´+y=t^2 , y(0)=0