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Solve the IVP y = 1+ y, y(0) = 1. On what maximum interval does the...
(Q3) Consider the equation: y′ = y1/3, y(0) = 0 . (a)Does the above IVP have...
(Q3) Consider the equation:
y′ = y1/3, y(0) = 0
. (a)Does the above IVP have any solution?
(b)Is the solution unique?
(c)Interpret your results in light of the theorem of existence
and uniqueness.
(Q3) Consider the equation: y' = y1/3, y(0) = 0 . (a)Does the above IVP have any solution? (b) Is the solution unique? (c)Interpret your results in light of the theorem of existence and uniqueness. (Q4) Solve the following IVP and find the interval of validity:...
Question 10 Incorrect Mark 0.00 out of 4.00 p Flag question Given the IVP y(0) =...
Question 10 Incorrect Mark 0.00 out of 4.00 p Flag question Given the IVP y(0) = 1 Without explicitly solving the ODE, indicate which of the following statements are true. Select one or more: a. The existence and uniqueness theorem guarantees the existence of a unique solution defined in an interval (h, h). b. The existence and uniqueness theorem guarantees the existence of a unique solution defined in an interval (1 - h. 1 + h). X c. The solution...
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
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...
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...
4. Consider the following initial value problem: y(0) = e. (a) Solve the IVP using the...
4. Consider the following initial value problem: y(0) = e. (a) Solve the IVP using the integrating factor method. (b) What is the largest interval on which its solution is guaranteed to uniquely exist? (c) The equation is also separable. Solve it again as a separable equation. Find the particular solution of this IVP. Does your answer agree with that of part (a)? 5 Find the general solution of the differential equation. Do not solve explicitly for y. 6,/Solve explicitly...
2. Indicate a rectangle (that is, an interval of t-values and an interval of y-values) in...
2. Indicate a rectangle (that is, an interval of t-values and an interval of y-values) in which the requirements of the theorem on existence and uniqueness are satisfied for the non-linear initial value problem dy 1 sin(t)y(ty 2y +4t - 8) = 0 dt with the given initial condition. If no such rectangle exists, explain why not. Do NOT solve the equation y(5) 5 (b) (c) y(1)4 (a) y(0) 3 = =
2. Indicate a rectangle (that is, an interval...
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...
2y 1. (9 points) Given the initial value problem y' = y (xo) = yo. Use...
2y 1. (9 points) Given the initial value problem y' = y (xo) = yo. Use the existence and uniqueness theorem to show that a) a unique solution exists on any interval where x, 60, b) no solution exists if y(0) = % 70, and c) an infinite number of solutions exist if y(0) = 0.
(Q3) Consider the equation:
y′ = y1/3, y(0) = 0
. (a)Does the above IVP have any solution?
(b)Is the solution unique?
(c)Interpret your results in light of the theorem of existence
and uniqueness.
(Q3) Consider the equation: y' = y1/3, y(0) = 0 . (a)Does the above IVP have any solution? (b) Is the solution unique? (c)Interpret your results in light of the theorem of existence and uniqueness. (Q4) Solve the following IVP and find the interval of validity:...
Question 10 Incorrect Mark 0.00 out of 4.00 p Flag question Given the IVP y(0) = 1 Without explicitly solving the ODE, indicate which of the following statements are true. Select one or more: a. The existence and uniqueness theorem guarantees the existence of a unique solution defined in an interval (h, h). b. The existence and uniqueness theorem guarantees the existence of a unique solution defined in an interval (1 - h. 1 + h). X c. The solution...
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...
4. Consider the following initial value problem: y(0) = e. (a) Solve the IVP using the integrating factor method. (b) What is the largest interval on which its solution is guaranteed to uniquely exist? (c) The equation is also separable. Solve it again as a separable equation. Find the particular solution of this IVP. Does your answer agree with that of part (a)? 5 Find the general solution of the differential equation. Do not solve explicitly for y. 6,/Solve explicitly...
2. Indicate a rectangle (that is, an interval of t-values and an interval of y-values) in which the requirements of the theorem on existence and uniqueness are satisfied for the non-linear initial value problem dy 1 sin(t)y(ty 2y +4t - 8) = 0 dt with the given initial condition. If no such rectangle exists, explain why not. Do NOT solve the equation y(5) 5 (b) (c) y(1)4 (a) y(0) 3 = =
2. Indicate a rectangle (that is, an interval...
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...
2y 1. (9 points) Given the initial value problem y' = y (xo) = yo. Use the existence and uniqueness theorem to show that a) a unique solution exists on any interval where x, 60, b) no solution exists if y(0) = % 70, and c) an infinite number of solutions exist if y(0) = 0.
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