Question

This is about differential equations. Consider the equation for a damped oscillator with no external force....

This is about differential equations.

Consider the equation for a damped oscillator with no external force. Then the equation becomes my’’ +by’ + ky = 0. Show that if y(t) is a solution to this equation, then cy(t) is a solution for any constant c. Also show that if y1(t) and y2(t) are solutions to this equation then c1y1(t) + c2y2(t) is also a solution for any c1 and c2.

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Consider the differential equation L[y] = y′′ + p(t)y′ + q(t)y = f(t) + g(t), and...
Consider the differential equation L[y] = y′′ + p(t)y′ + q(t)y = f(t) + g(t), and suppose L[yf] = f(t) and L[yg] = g(t). Explain why yp = yf + yg is a solution to L[y] = f + g. Suppose y and y ̃ are both solutions to L[y] = f + g, and suppose {y1, y2} is a fundamental set of solutions to the homogeneous equation L[y] = 0. Explain why y = C1y1 + C2y2 + yf...
Consider the equation of motion of the damped oscillator. Determine the rate of change of mechanical...
Consider the equation of motion of the damped oscillator. Determine the rate of change of mechanical energy E by direct derivation and, with the aid of the equation of motion, show that dE/dt is the rate at which energy is dissipated by the frictional force.
The indicated function y1(x) is a solution of the given differential equation. Use reduction of order...
The indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, y2 = y1(x) e−∫P(x) dx y 2 1 (x) dx (5) as instructed, to find a second solution y2(x). y'' + 100y = 0; y1 = cos 10x I've gotten to the point all the way to where y2 = u y1, but my integral is wrong for some reason This was my answer y2= c1((sin(20x)+20x)cos10x)/40 + c2(cos(10x))
a) The homogeneous and particular solutions of the differential equation ay'' + by' + cy =...
a) The homogeneous and particular solutions of the differential equation ay'' + by' + cy = f(x) are, respectively, C1exp(x)+C2exp(-x) and 3x^3. Give the complete solution y(x) of the differential equation. b) If the force f(x) in the equation given in a) is instead f(x) = f1(x) + f2(x) + f3(x), where f1(x), f2(x), and f3(x) are generic forces, what would be the particular solution? c) The homogeneous solution of a forced oscillator is cos(t) + sin(t), what is the...
The position x(t), of a damped oscillator with forcing satisfies the ordinary differential equation , i)...
The position x(t), of a damped oscillator with forcing satisfies the ordinary differential equation , i) where f(t) denotes the forcing on the oscillator. (i) If x(0) = 0, dx dt (0) = 1, f(t) = 4t and the Laplace transform of x(t) is denoted X(s) = L[x(t)], then show that X(s) = 1 /(s + 2)^2 + 4 /s^2 (s + 2)^2 ii) Hence find x(t)
Choose the correct answers If y1 and y2 are two solutions of a nonhomogeneous equation ayjj+...
Choose the correct answers If y1 and y2 are two solutions of a nonhomogeneous equation ayjj+ byj+ cy =f (x), then their difference is a solution of the equation ayjj+ byj+ cy = 0. If f (x) is continuous everywhere, then there exists a unique solution to the following initial value problem.                                   f (x)yj= y,   y(0) = 0 The differential equation yjj + t2yj − y = 3 is linear. There is a solution to the ODE yjj+3yj+y...
Consider the differential equation x^2y′′ − 3xy′ − 5y = 0. Note that this is not...
Consider the differential equation x^2y′′ − 3xy′ − 5y = 0. Note that this is not a constant coefficient differential equation, but it is linear. The theory of linear differential equations states that the dimension of the space of all homogeneous solutions equals the order of the differential equation, so that a fundamental solution set for this equation should have two linearly fundamental solutions. • Assume that y = x^r is a solution. Find the resulting characteristic equation for r....
Consider the autonomous differential equation dy/ dt = f(y) and suppose that y1 is a critical...
Consider the autonomous differential equation dy/ dt = f(y) and suppose that y1 is a critical point, i.e., f(y1) = 0. Show that the constant equilibrium solution y = y1 is asymptotically stable if f 0 (y1) < 0 and unstable if f 0 (y1) > 0.
Consider the driven damped harmonic oscillator m(d^2x/dt^2)+b(dx/dt)+kx = F(t) with driving force F(t) = FoSin(wt). Consider...
Consider the driven damped harmonic oscillator m(d^2x/dt^2)+b(dx/dt)+kx = F(t) with driving force F(t) = FoSin(wt). Consider the overdamped case (b/2m)^2 < k/m a. Find the steady state solution. b. Find the solution with initial conditions x(0)=0, x'(0)=0. c. Use a plotting program to plot your solution for m=1, k=0.1, b=1, Fo=0.25, and w=0.5.
Consider the differential equation: 66t^2y''+12t(t-11)y'-12(t-11)y=5t^3, . You can verify that y1 = 5t and y2 =...
Consider the differential equation: 66t^2y''+12t(t-11)y'-12(t-11)y=5t^3, . You can verify that y1 = 5t and y2 = 4te^(-2t/11)satisfy the corresponding homogeneous equation. The Wronskian W between y1 and y2 is W(t) = (-40/11)t^2e^((-2t)/11) Apply variation of parameters to find a particular solution. yp = ?????
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT