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Please show steps. Given 3 dy/dx + 2xy^2 = 5x^2 - x + 1, where y(0)...
Please show steps.
Given 3 dy/dx + 2xy^2 = 5x^2 - x + 1, where y(0) = 5 and using a step size of dx = 1, the value of y(1) using Euler's method is most nearly 5.333 1.010 -0.499 17.822 Given 3 dy/dx + 2xy^2 = 5x^2 - x + 1, where y(0) = 5 and using a step size of dx = 1, the value of y(1) using Runge-Kutta 4^th order method is most nearly 5.333 1.010 -0.499...
(e) Consider the Runge-Kutta method in solving the following first order ODE: dy First, using Taylor series expansion, we have the following approximation of y evaluated at the time step n+1 as a...
(e) Consider the Runge-Kutta method in solving the following first order ODE: dy First, using Taylor series expansion, we have the following approximation of y evaluated at the time step n+1 as a function of y at the time step n: where h is the size of the time step. The fourth order Runge-Kutta method assumes the following form where the following approximations can be made at various iterations: )sh+รู้: ,f(t.ta, ),. Note that the first term is evaluated at...
Solve the initial value problem y' = x(y - x), y(2) = 3 in the interval...
Solve the initial value problem y' = x(y - x), y(2) = 3 in the interval [2,3] using Runge Kutta fourth order with step size of h = 0.2.
(16 marks) Consider the initial value problem (a) Without using pre-built commands write an m-file function...
(16 marks) Consider the initial value problem (a) Without using pre-built commands write an m-file function that uses the fourth-order Runge-Kutta method to estimate the value of y(n) for a given value n and a given step size h (b) Use the m-file function built in part (a) to compute an estimate of y(2) using step size h = 0.5 and h = 0.25. Fron these two estimates, approximate the step size needed to estimate y(2) correct to 4 decimal...
The Program for the code should be matlab 5. [25 pointsl Given the initial value problem...
The Program for the code should be matlab
5. [25 pointsl Given the initial value problem with the initial conditions y(0) 2 and y'(0)10, (a) Solve analytically to obtain the exact solution y(x) (b) Solve numerically using the forward Euler, backward Euler, and fourth-order Runge Kutta methods. Please implement all three methods yourselves do not use any built- in integrators (i.e., ode45)). Integrate over 0 3 r < 4, and compare the methods with the exact solution. (For example, using...
Can you help me with this problem? This has to be done using Matlab and solving with runge-katta, euler method, and the built in function ode45
Consider a cylindrical storage tank with surface area A which contains a liquid at depth y:At time t = 0, the tank is empty (y = 0 when t = 0). Liquid is supplied to the tank at a sinusoidal
rate Qin =3Qsin2
(t) and withdrawn from the tank as:
𝑄𝑜𝑢𝑡 = 3(𝑦 − 𝑦𝑜𝑢𝑡)
1.5
if 𝑦 > 𝑦𝑜𝑢𝑡
𝑄𝑜𝑢𝑡 = 0 if 𝑦 ≤ 𝑦𝑜𝑢𝑡 Please note that both 𝑄𝑖𝑛 and 𝑄𝑜𝑢𝑡 have units m3
/h. The mass...
2. Consider the following first-order ODE from x = 0 to x = 2.4 with y(0)...
2. Consider the following first-order ODE from x = 0 to x = 2.4 with y(0) = 2. (a) solving with Euler's explicit method using h=0.6 (b) solving with midpoint method using h= 0.6 (c) solving with classical fourth-order Runge-Kutta method using h = 0.6. Plot the x-y curve according to your solution for both (a) and (b).
4. (25 points) Solve the following ODE using classical 4th-order Runge- Kutta method within the domain...
4. (25 points) Solve the following ODE using classical 4th-order Runge- Kutta method within the domain of x = 0 to x= 2 with step size h = 1: dy 3 dr=y+ 6x3 dx The initial condition is y(0) = 1. If the analytical solution of the ODE is y = 21.97x - 5.15; calculate the error between true solution and numerical solution at y(1) and y(2).
Question 9 10 pts Given the ODE, initial condition, and step size: dx + 2x3 =...
Question 9 10 pts Given the ODE, initial condition, and step size: dx + 2x3 = e-3t x(3) = 1 and At = 0.5. We intend to find x(t=3.5) using the Runge Kutta 4 method. Find all the slopes and the required increment function for this step. Input your answers to 4 decimal places increment function =
ay=-ay, a>0, y(0) = yo. dx (1) For this model problem, derive the maximum step size...
ay=-ay, a>0, y(0) = yo. dx (1) For this model problem, derive the maximum step size for which Heun’s method (i.e. 2nd-order Runge-Kutta, with az = 1/2) remains numerically stable.
Please show steps.
Given 3 dy/dx + 2xy^2 = 5x^2 - x + 1, where y(0) = 5 and using a step size of dx = 1, the value of y(1) using Euler's method is most nearly 5.333 1.010 -0.499 17.822 Given 3 dy/dx + 2xy^2 = 5x^2 - x + 1, where y(0) = 5 and using a step size of dx = 1, the value of y(1) using Runge-Kutta 4^th order method is most nearly 5.333 1.010 -0.499...
(e) Consider the Runge-Kutta method in solving the following first order ODE: dy First, using Taylor series expansion, we have the following approximation of y evaluated at the time step n+1 as a function of y at the time step n: where h is the size of the time step. The fourth order Runge-Kutta method assumes the following form where the following approximations can be made at various iterations: )sh+รู้: ,f(t.ta, ),. Note that the first term is evaluated at...
Solve the initial value problem y' = x(y - x), y(2) = 3 in the interval [2,3] using Runge Kutta fourth order with step size of h = 0.2.
(16 marks) Consider the initial value problem (a) Without using pre-built commands write an m-file function that uses the fourth-order Runge-Kutta method to estimate the value of y(n) for a given value n and a given step size h (b) Use the m-file function built in part (a) to compute an estimate of y(2) using step size h = 0.5 and h = 0.25. Fron these two estimates, approximate the step size needed to estimate y(2) correct to 4 decimal...
The Program for the code should be matlab
5. [25 pointsl Given the initial value problem with the initial conditions y(0) 2 and y'(0)10, (a) Solve analytically to obtain the exact solution y(x) (b) Solve numerically using the forward Euler, backward Euler, and fourth-order Runge Kutta methods. Please implement all three methods yourselves do not use any built- in integrators (i.e., ode45)). Integrate over 0 3 r < 4, and compare the methods with the exact solution. (For example, using...
2. Consider the following first-order ODE from x = 0 to x = 2.4 with y(0) = 2. (a) solving with Euler's explicit method using h=0.6 (b) solving with midpoint method using h= 0.6 (c) solving with classical fourth-order Runge-Kutta method using h = 0.6. Plot the x-y curve according to your solution for both (a) and (b).
4. (25 points) Solve the following ODE using classical 4th-order Runge- Kutta method within the domain of x = 0 to x= 2 with step size h = 1: dy 3 dr=y+ 6x3 dx The initial condition is y(0) = 1. If the analytical solution of the ODE is y = 21.97x - 5.15; calculate the error between true solution and numerical solution at y(1) and y(2).
Question 9 10 pts Given the ODE, initial condition, and step size: dx + 2x3 = e-3t x(3) = 1 and At = 0.5. We intend to find x(t=3.5) using the Runge Kutta 4 method. Find all the slopes and the required increment function for this step. Input your answers to 4 decimal places increment function =
ay=-ay, a>0, y(0) = yo. dx (1) For this model problem, derive the maximum step size for which Heun’s method (i.e. 2nd-order Runge-Kutta, with az = 1/2) remains numerically stable.
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