HELP Initial Value Problem Question-Differential Equations and Euler's Method

[spazticbutter]

URGENT HELP! Initial Value Problem Question-Differential Equations and Euler's Method

Homework Statement

This is just an extension of an earlier thread. I see now that they want me to use Euler's method so it might change the way I do the problem.
The problem wants me to solve the initial value problem (for y I presume) and then use it with Euler's method to approximate the solution at t = 0.5,1,1.5,2,and 2.5.

Homework Equations

dy/dt = 5 - 3(y^(1/2)) y(0) = 2

The Attempt at a Solution

I took the equation as separable and took the integral of 1/(5-3(y^(1/2))) and found
(-10/9)ln l 5 - 3(y^(1/2)) l + (2/9)(5 - 3(y^(1/2))) = t + C. I don't see how I can solve for y from here. Did I perhaps do my integration wrong? Is there another way to solve this IVP without separation? Or perhaps I don't need to solve for y?
I know that I can solve for C using the given initial value of y but I don't see how this would help me. I know how to do Euler's method so you don't need to explain how to do that unless I have to do it in some special way to figure this problem out
Any help would be appreciated

 

[mighty2000]

!Dear student,

Thank you for reaching out for help with your initial value problem question. I understand that you are struggling with using Euler's method to approximate the solution at certain values of t. Let me try to guide you through the process.

First, you are correct in using Euler's method for this problem. It is a numerical method used to approximate the solution of a differential equation, and it is particularly useful when the equation is not easily solvable by hand.

To start, you will need to find the general solution to the differential equation, which you have correctly identified as separable. However, I believe there may be a mistake in your integration. The correct integral is (-10/9)ln |5 - 3(y^(1/2))| + (2/9)(5 - 3(y^(1/2))) + C = t + C. You can check this by differentiating the left side and verifying that it gives you the original differential equation.

Next, you will need to use the initial condition given (y(0) = 2) to solve for the value of C. This will give you the particular solution to the differential equation.

Now, to use Euler's method, you will need to choose a step size, h, which represents the distance between the values of t that you want to approximate the solution for. In this case, you are given specific values of t (0.5, 1, 1.5, 2, and 2.5), so you can choose a small enough step size to ensure accuracy (such as h = 0.1).

Finally, you can use the formula for Euler's method to approximate the solution at each of the given values of t. The formula is y(n+1) = y(n) + hf(t(n), y(n)), where y(n) represents the approximation of the solution at t = n*h, and f(t(n), y(n)) represents the value of the differential equation at t = n*h and y = y(n).

I hope this helps you to better understand how to approach this problem. Please let me know if you have any further questions or need clarification on any of the steps. Good luck!