Given a second order ode m y’’ + c y’ + k y = 0 with m, c...

Given a second order ode m y’’ + c y’ + k y = 0 with m, c and k all positive. (like a mass‐spring system with damping) Argue that the solution will always be damped; the exponential portion can never be positive regardless of the particular m, c and k.

Expert Solution

Colby Messinger answered 1 year ago

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.

Consider the following second-order ODE: (d^2 y)/(dx^2 )+2 dy/dx+2y=0 from x = 0 to x =...

Consider the following second-order ODE: (d^2 y)/(dx^2 )+2 dy/dx+2y=0 from x = 0 to x = 1.6 with y(0) = -1 and dy/dx(0) = 0.2. Solve with Euler’s explicit method using h = 0.4. Plot the x-y curve according to your solution.

Use ‘Reduction of Order’ to find a second solution y2 to the given ODEs: (a) y′′+2y′+y=0,...

Use ‘Reduction of Order’ to find a second solution y2 to the given ODEs: (a) y′′+2y′+y=0, y1 =xe−x (b) y′′+9y=0, y1 =sin3x (c) x2y′′+2xy′−6y=0, y1 =x2 (d) xy′′ +y′ =0, y1 =lnx

Consider the following first-order ODE dy/dx=x^2/y from x = 0 to x = 2.4 with y(0)...

Consider the following first-order ODE dy/dx=x^2/y from x = 0 to x = 2.4 with y(0) = 2. (a) solving with Euler’s explicit method using h = 0.6 (b) solving with midpoint method using h = 0.6 (c) solving with classical fourth-order Runge-Kutta method using h = 0.6. Plot the x-y curve according to your solution for both (a) and (b).

solve for the second order initial value problem= y"-6y+8y=0 y(0)=3, y'(0)=10

Lets say I have a second order ODE: mx''+kx+bx'=kz+bz' ( an EOM where m=3000, b=2000, k=50000,...

Lets say I have a second order ODE: mx''+kx+bx'=kz+bz' ( an EOM where m=3000, b=2000, k=50000, and finally z=.01sin(3.5t)). In Matlab, how would I use the Matlab command ode23 to solve this ode assuming zero initial conditions, range :0-20 seconds.

Solve the linear second-order ODE for each case of b. Find constants using the given initial...

Solve the linear second-order ODE for each case of b. Find constants using the given initial conditions. y(0)=1, y'(0)=0 y''+by'+16y=0 b=0 b=2 b=8 b=10 say b represents damping constant. What is the effect of damping on the motion of a mass?

Find a power series solution of this ODE: y''+x^2y'=0 ; y(0)=1 y'(0)= 2

3. Consider4 the homogenous linear second order differential equation y′′ − 2y′ + y = 0...

3. Consider4 the homogenous linear second order differential equation y′′ − 2y′ + y = 0 (⋆) (a) Verify that the function y = e^x is a solution of equation (⋆) on the interval (−∞, ∞). (b) Verify that the function y = xex is a solution of equation (⋆) on the interval (−∞, ∞). (c) Verify that y = 7e^x + (5xe)^x is a solution of equation (⋆) on the interval (−∞, ∞). (d) Assume that c and d...

For the following ODE 4. xy'' +y' -xy = 0; x0 = 0 a) What are...

For the following ODE 4. xy'' +y' -xy = 0; x0 = 0 a) What are the points of singularity for the problem? b) Does this ODE hold a general power series solution at the specific x0? Justify your answer, i) if your answer is yes, the proceed as follows: Compute the radius of analyticity and report the corresponding interval, and Identify the recursion formula for the power series coefficient around x0, and write the corresponding solution with at least...

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