Question

In: Advanced Math

3. Consider the IVP: dy =ty^1/3; y(0)=0,t≥0. dt Both y(t) = 0, (the equilibrium solution) and...

3. Consider the IVP:

dy =ty^1/3; y(0)=0,t≥0. dt

Both y(t) = 0, (the equilibrium solution) and y(t) = ?(1/3t^2?)^3/2 are solutions to this IVP.

  1. (a) Show that the trivial solution satisfies the IVP by first verifying that it satisfies the initial condition and then

    verifying that it satisfies the differential equation.

  2. (b) Show that the other solution satisfies the IVP again by first verifying it satisfies the initial condition and then verifying that it satisfies the differential equation.

  3. (c) Explain in your own words why having these two solutions does not violate the existence and uniqueness theorem we discussed in class.

Solutions

Expert Solution

Colby Messinger answered 5 months ago
Next > < Previous

Related Solutions

Consider the following initial value problem dy/dt = 3 − 2*t − 0.5*y, y (0) =...
Consider the following initial value problem dy/dt = 3 − 2*t − 0.5*y, y (0) = 1 We would like to find an approximation solution with the step size h = 0.05. What is the approximation of y(0.1)?
for y(t) function ty'' - ty' + ty = 0, y(0)= 0 , y'(0)= 1 solve...
for y(t) function ty'' - ty' + ty = 0, y(0)= 0 , y'(0)= 1 solve this initial value problem by using Laplace Transform. (The equation could have been given such as "y'' - y' + y = 0" but it is not. Please, be careful and solve this question step by step.) )
Consider the system modeled by the differential equation dy/dt - y = t with initial condition y(0) = 1
Consider the system modeled by the differential equation                               dy/dt - y = t    with initial condition y(0) = 1 the exact solution is given by y(t) = 2et − t − 1   Note, the differential equation dy/dt - y =t can be written as                                               dy/dt = t + y using Euler’s approximation of dy/dt = (y(t + Dt) – y(t))/ Dt                               (y(t + Dt) – y(t))/ Dt = (t + y)                                y(t + Dt) =...
dy/dt +2y=t-1. ivp y0 =1 solve differntial equation
dy/dt +2y=t-1. ivp y0 =1 solve differntial equation
Use a LaPlace transform to solve d^2x/dt^2+dx/dt+dy/dt=0 d^2y/dt^2+dy/dt-4dy/dt=0 x(0)=1,x'(0)=0 y(0)=-1,y'(0)=5
Use a LaPlace transform to solve d^2x/dt^2+dx/dt+dy/dt=0 d^2y/dt^2+dy/dt-4dy/dt=0 x(0)=1,x'(0)=0 y(0)=-1,y'(0)=5
6. Consider the initial value problem dy/ dt = t ^2 y , y(1) = 1...
6. Consider the initial value problem dy/ dt = t ^2 y , y(1) = 1 . (a) Use Euler’s method (by hand) to approximate the solution y(t) at t = 2 using ∆t = 1, ∆t = 1/2 and ∆t = 1/4 (use a calculator to approximate the answer for the smallest ∆t). Report your results in a table listing ∆t in one column, and the corresponding approximation of y(2) in the other. (b) The direction field of dy/dt...
Solve the following initial value problems (1) dy/dt = t + y y(0) = 1 so...
Solve the following initial value problems (1) dy/dt = t + y y(0) = 1 so y(t) = (2)  dy/dt = ty y(0) = 1 so y(t) =
Consider the IVP t2y''−(t2 + 2)y' + (t + 2)y = t3 with y(1) = 0...
Consider the IVP t2y''−(t2 + 2)y' + (t + 2)y = t3 with y(1) = 0 and y'(1) = 0. • One function in the fundamental set of solutions is y1(t) = t. Find the second function y2(t) by setting y2(t) = w(t)y1(t) for w(t) to be determined. • Find the solution of the IVP
Consider the IVP t2y''−(t2 + 2)y' + (t + 2)y = t3 with y(1) = 0...
Consider the IVP t2y''−(t2 + 2)y' + (t + 2)y = t3 with y(1) = 0 and y'(1) = 0. • One function in the fundamental set of solutions is y1(t) = t. Find the second function y2(t) by setting y2(t) = w(t)y1(t) for w(t) to be determined. • Find the solution of the IVP
1) . Solve the IVP: y^''+6y^'+5y=0, y(0)=1, y^' (0)=3 2. Find the general solution to each...
1) . Solve the IVP: y^''+6y^'+5y=0, y(0)=1, y^' (0)=3 2. Find the general solution to each of the following: a) y^''+2y^'+5y=e^2x b) y^''+2x/(x^2+1) y'=x c) y^''+4y=1/(sin⁡(2x)) (use variation of parameters)
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT