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Problem 2. Proof the Initial Value Theorem: lim q(t) = lim sQ(s). E $ 0 Hint:...
Solve the initial value problem honhomogeneous equation: LQ"+RQ+ Q = E. coswt initial Conditions: SQ (0)=Qe,...
Solve the initial value problem honhomogeneous equation: LQ"+RQ+ Q = E. coswt initial Conditions: SQ (0)=Qe, Q (0) = Q! as follows: + First solve the associated homogeneous lequation LQ"+RQ'TEQ =0 by using the characteristic equation Irt art &r=0 to obtain three types of solutions 2 Next show how to find a particular solution Qp to the non homogeneous equation by showing Oplt) = t-Lw²coswt twR sin (wt) ( - Lw²) 4 w²R Eo Show in detail that you can...
(complete the proof. Hint: Use the Squeeze Theorem to show that lima = L.) 3- For...
(complete the proof. Hint: Use the Squeeze Theorem to show that lima = L.) 3- For all ne N, let an = Let S = {a, neN). 3-1) Use the fact that lim 0 and the result of Exercise 1 to show that OES'. 3-2) Use the result of Exercise 2 to show that S - {0}. 4- Prove that
solve the initial value problem. Aft) Use the method of Laplace transforms and the accompanying proof...
solve the initial value problem. Aft) Use the method of Laplace transforms and the accompanying proof results y"+ @y' +8y=f(t): y(O)=0. y' (O)=0 Here,f(t) is the periodic function defined in the graph to the right. Q Click the icon to review the results of a proof. Square wave Choose the correct answer to the initial value problem below. y(t)= (1-e-232 3 •B. y(t)= (1-e-2t-4n)Puſt - 4n) n=0 • C. y(t)= (-1) (1 - e -2(1-21) jult - 2n) 0 3...
2. Problem: Given Q(x)=2(2-1) . Give a step-by-step(δ Proof to prove that: lim QCx) 1. ing...
2. Problem: Given Q(x)=2(2-1) . Give a step-by-step(δ Proof to prove that: lim QCx) 1. ing the ε-δ definition you are using for this problem in terms of the formula of Q(x) and limit value
Problem 5: Consider the initial value Dirichlet problem ur(t, x) - 2uzz(t, x) = e, (t,...
Problem 5: Consider the initial value Dirichlet problem ur(t, x) - 2uzz(t, x) = e, (t, x) € (0, +00), u(0,x) = 1, u(t,0) = e. For the unique solution u(x, t) find the following limit as a function of t: lim u(x, t).
. Problem 3 a) (2 points) What is the initial value of time function f(t) corresponding...
. Problem 3 a) (2 points) What is the initial value of time function f(t) corresponding to the one sided Laplace Transform F(s) = 365+1096+4) (.e. f(t) is causal) lim f(t) = 00 1-0 limf(t) = 1. 10x4 lim f(t) = 0 • limf(t) cannot computed since sF(s) is not analytic. None of these choices is correct. -0 . t-0 t+0 . b) (2 points) What is the final value of time function f(t) corresponding to the one 40 sided...
For each initial value problem, does Picards's theorem apply? If so, determine if it guarantees that...
For each initial value problem, does Picards's theorem apply? If
so, determine if it guarantees that a solutio exists and is
unique.
Theorem (Picard). Consider the initial value problem dy = f(t,y), dt (IVP) y(to) = Yo- (a) Existence: If f(t,y) is continuous in an open rectangle R = {(t,y) |a<t < b, c < y < d} and (to, Yo) belongs in R, then there exist h > 0 and a solution y = y(t) of (IVP) defined in...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint:...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Use the method of Laplace transforms and the accompanying proof results to solve the initial value...
Use the method of Laplace transforms and the accompanying proof results to solve the initial value problem. Af(t) 4 y"' + 3y' + 2y = f(t); y(0) = 0, y'(0) = 0 Here, f(t) is the periodic function defined in the graph to the right. Click the icon to review the results of a proof. 0 16 8 12 Square wave Choose the correct answer to the initial value problem below. O A. y(t) = 2 (-1^(1 - e -...
Problem 2. (a) Solve the initial value problem I y' + 2y = g(t), 1 y(0)...
Problem 2. (a) Solve the initial value problem I y' + 2y = g(t), 1 y(0) = 0, where where | 1 if t < 1, g(t) = { 10 if t > 1 (t) = { for all t. Is this solution unique for all time? Is it unique for any time? Does this contradict the existence and uniqueness theorem? Explain. (b) If the initial condition y(0) = 0 were replaced with y(1) = 0, would there necessarily be...
Solve the initial value problem honhomogeneous equation: LQ"+RQ+ Q = E. coswt initial Conditions: SQ (0)=Qe, Q (0) = Q! as follows: + First solve the associated homogeneous lequation LQ"+RQ'TEQ =0 by using the characteristic equation Irt art &r=0 to obtain three types of solutions 2 Next show how to find a particular solution Qp to the non homogeneous equation by showing Oplt) = t-Lw²coswt twR sin (wt) ( - Lw²) 4 w²R Eo Show in detail that you can...
(complete the proof. Hint: Use the Squeeze Theorem to show that lima = L.) 3- For all ne N, let an = Let S = {a, neN). 3-1) Use the fact that lim 0 and the result of Exercise 1 to show that OES'. 3-2) Use the result of Exercise 2 to show that S - {0}. 4- Prove that
solve the initial value problem. Aft) Use the method of Laplace transforms and the accompanying proof results y"+ @y' +8y=f(t): y(O)=0. y' (O)=0 Here,f(t) is the periodic function defined in the graph to the right. Q Click the icon to review the results of a proof. Square wave Choose the correct answer to the initial value problem below. y(t)= (1-e-232 3 •B. y(t)= (1-e-2t-4n)Puſt - 4n) n=0 • C. y(t)= (-1) (1 - e -2(1-21) jult - 2n) 0 3...
2. Problem: Given Q(x)=2(2-1) . Give a step-by-step(δ Proof to prove that: lim QCx) 1. ing the ε-δ definition you are using for this problem in terms of the formula of Q(x) and limit value
Problem 5: Consider the initial value Dirichlet problem ur(t, x) - 2uzz(t, x) = e, (t, x) € (0, +00), u(0,x) = 1, u(t,0) = e. For the unique solution u(x, t) find the following limit as a function of t: lim u(x, t).
. Problem 3 a) (2 points) What is the initial value of time function f(t) corresponding to the one sided Laplace Transform F(s) = 365+1096+4) (.e. f(t) is causal) lim f(t) = 00 1-0 limf(t) = 1. 10x4 lim f(t) = 0 • limf(t) cannot computed since sF(s) is not analytic. None of these choices is correct. -0 . t-0 t+0 . b) (2 points) What is the final value of time function f(t) corresponding to the one 40 sided...
For each initial value problem, does Picards's theorem apply? If
so, determine if it guarantees that a solutio exists and is
unique.
Theorem (Picard). Consider the initial value problem dy = f(t,y), dt (IVP) y(to) = Yo- (a) Existence: If f(t,y) is continuous in an open rectangle R = {(t,y) |a<t < b, c < y < d} and (to, Yo) belongs in R, then there exist h > 0 and a solution y = y(t) of (IVP) defined in...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Use the method of Laplace transforms and the accompanying proof results to solve the initial value problem. Af(t) 4 y"' + 3y' + 2y = f(t); y(0) = 0, y'(0) = 0 Here, f(t) is the periodic function defined in the graph to the right. Click the icon to review the results of a proof. 0 16 8 12 Square wave Choose the correct answer to the initial value problem below. O A. y(t) = 2 (-1^(1 - e -...
Problem 2. (a) Solve the initial value problem I y' + 2y = g(t), 1 y(0) = 0, where where | 1 if t < 1, g(t) = { 10 if t > 1 (t) = { for all t. Is this solution unique for all time? Is it unique for any time? Does this contradict the existence and uniqueness theorem? Explain. (b) If the initial condition y(0) = 0 were replaced with y(1) = 0, would there necessarily be...
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