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3. (Existence/uniqueness theorem, Strogatz 6.2): Consider the systenm (a) Show by substitution th...
3. (Existence/uniqueness theorem, Strogatz 6.2): Consider the system (a) Show by substitution tha...
3. (Existence/uniqueness theorem, Strogatz 6.2): Consider the system (a) Show by substitution that (t) sin, () cost is an exact solution. (b) Now consider another solution, with initial condition a(0)-1/2, y(0) 0. Without doing any work, explain why this solution must satisfy y2 < 1 for all t < oo. For the systems in problems 4-7, find the fixed points, linearize about them, classify their stability, draw their local trajectories, and try to fill in the full phase portrait.
3....
Consider the initial value problem. Apply the Fundamental Existence and Uniqueness Theorem to show that a...
Consider the initial value problem.
Apply the Fundamental Existence and Uniqueness Theorem to show
that a solution exists. Solve the IVP using your favorite method.
What is the domain of definition of the solution function?
y(0) = 1.
Consider the IVP, 1. Apply the Fundamental Existence and Uniqueness Theorem to show that a solution...
Consider the IVP, 1. Apply the Fundamental Existence and Uniqueness Theorem to show that a solution exists. 2. Use the Runge-Kutta method with various step-sizes to estimate the maximum t-value, , for which the solution is defined on the interval . Include a few representative graphs with your submission, and the lists of points. 3. Find the exact solution to the IVP and solve for analytically. How close was your approximation from the previous question? 4. The Runge-Kutta method continues...
given ivp y' = (2y)/x, y(x0) = y0 using the existence and uniqueness theorem show that...
given ivp y' = (2y)/x, y(x0) = y0 using the existence and uniqueness theorem show that a unique solution exists on any interval where x0 does not equal 0, no solution exists if y(0) = y0 does not equal 0, and and infinite number of solutions exist if y(0) = 0
According to the Existence and Uniqueness theorem, the differential equation (t−5)y′+ysin(t)=5t ...
According to the Existence and Uniqueness theorem, the differential equation (t−5)y′+ysin(t)=5t necessarily has a unique solution on the interval 0<t≤5. TRUE FALSE A numerical method is said to converge if its approximate solution values for a differential equation y′=f(t,y), y1,y2,...,yn, approach the true solution values ϕ(t1),ϕ(t2),...,ϕ(tn), as the stepsize h→∞. TRUE FALSE If a numerical method has a global truncation error that is proportional to the nth power of the stepsize, then it is called an nth order method. TRUE...
6. (2 pts) Consider the following initial value problem: y' = (t + y)?y2 + sin(yº)...
6. (2 pts) Consider the following initial value problem: y' = (t + y)?y2 + sin(yº) + yety, y(0) = 0. This initial value problem satisfies the existence and uniqueness theorem criteria using interval (-0, 0) for both thet and y variables, and hence has a unique solutoin. Find this unique solution. Hint: None of the techniques we've learned for explicitly solving will work. Instead, try plugging the initial condition into the differential equation and think about what that tells...
Consider differential equation (x - 1)y" – xy' + y = 0. a). Show that yi...
Consider differential equation (x - 1)y" – xy' + y = 0. a). Show that yi = el is a solution of this equation. Use the method of reduction of order to find second linearly independent solution y2 of this equation. (2P.) b). Find solution of the initial value problem (1P.) y(1) = 0, y'(1) = 1. c). Find solution of the initial value problem (1P.) y(1) = 0, y'(1) = 0. d). Does your answer in b) and c)...
4. Consider the differential equation with initial condition r(0) = 0 (a) What does the existence...
4. Consider the differential equation with initial condition r(0) = 0 (a) What does the existence and uniqueness theorem tell you about the solution to this IVP? (10 points) (b) Use separation of variables to find the solution for the IVP r(to) = Io for to +0. (5 points) (c) Are the solutions to b) unique? (5 points) (d) Sketch solutions for Xo = --1,0,1 and to = 1 and show that for all to and to the solution goes...
Alpha=9 beta=3 yazarsin 1. Consider the following initial-value problem. y' = e(1+B)t ln(1 + y2), 0<t<1...
Alpha=9 beta=3 yazarsin
1. Consider the following initial-value problem. y' = e(1+B)t ln(1 + y2), 0<t<1 y (0) = a +1 a) ( 15p.) Determine the existence and uniqueness of the solution. b) ( 15p.) Use Euler's method with h = 0.25 to approximate the solution at t = 0.5. {"
Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system...
Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system t' =-y+px(x² + y) (4) y = 1+ y(x² + y2), where is a parameter. Obviously, the origin x* = (0,0) is a fixed point of (4). (e) The solution of the ODE for o(t) is obvious - the angle o increases at a constant rate. Without solving the ODE for r(t), explain how r(t) behaves when t o in the cases H<0,1 =...
3. (Existence/uniqueness theorem, Strogatz 6.2): Consider the system (a) Show by substitution that (t) sin, () cost is an exact solution. (b) Now consider another solution, with initial condition a(0)-1/2, y(0) 0. Without doing any work, explain why this solution must satisfy y2 < 1 for all t < oo. For the systems in problems 4-7, find the fixed points, linearize about them, classify their stability, draw their local trajectories, and try to fill in the full phase portrait.
3....
Consider the initial value problem.
Apply the Fundamental Existence and Uniqueness Theorem to show
that a solution exists. Solve the IVP using your favorite method.
What is the domain of definition of the solution function?
y(0) = 1.
6. (2 pts) Consider the following initial value problem: y' = (t + y)?y2 + sin(yº) + yety, y(0) = 0. This initial value problem satisfies the existence and uniqueness theorem criteria using interval (-0, 0) for both thet and y variables, and hence has a unique solutoin. Find this unique solution. Hint: None of the techniques we've learned for explicitly solving will work. Instead, try plugging the initial condition into the differential equation and think about what that tells...
Consider differential equation (x - 1)y" – xy' + y = 0. a). Show that yi = el is a solution of this equation. Use the method of reduction of order to find second linearly independent solution y2 of this equation. (2P.) b). Find solution of the initial value problem (1P.) y(1) = 0, y'(1) = 1. c). Find solution of the initial value problem (1P.) y(1) = 0, y'(1) = 0. d). Does your answer in b) and c)...
4. Consider the differential equation with initial condition r(0) = 0 (a) What does the existence and uniqueness theorem tell you about the solution to this IVP? (10 points) (b) Use separation of variables to find the solution for the IVP r(to) = Io for to +0. (5 points) (c) Are the solutions to b) unique? (5 points) (d) Sketch solutions for Xo = --1,0,1 and to = 1 and show that for all to and to the solution goes...
Alpha=9 beta=3 yazarsin
1. Consider the following initial-value problem. y' = e(1+B)t ln(1 + y2), 0<t<1 y (0) = a +1 a) ( 15p.) Determine the existence and uniqueness of the solution. b) ( 15p.) Use Euler's method with h = 0.25 to approximate the solution at t = 0.5. {"
Problem 3. Linearization of a nonlinear system at a non-hyperbolic fixed point] Consider the nonlinear system t' =-y+px(x² + y) (4) y = 1+ y(x² + y2), where is a parameter. Obviously, the origin x* = (0,0) is a fixed point of (4). (e) The solution of the ODE for o(t) is obvious - the angle o increases at a constant rate. Without solving the ODE for r(t), explain how r(t) behaves when t o in the cases H<0,1 =...
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