Nonlinear ODE - dy/dt=F(y(t),x(t))

[halycos]

I'm having issues approaching this problem. I need to solve for

Homework Statement

Given the following equation, I need to find the max change in x(t) as y(t) changes, given bounds $y_{max}$ and $y_{min}$.

$\frac{dy}{dt} + a \sqrt(y(t)) = b x(t)$

Homework Equations

All ODE methods, MATLAB, or any method of solving, but not linearizing the equation

The Attempt at a Solution

I know an attempt would be nice, but I have looked at this for hours with no progress. I was thinking that it was close enough to use an integrating factor but I can't figure out how to handle integrating

$\frac{1}{-a \sqrt(y(t)) + b x(t) } \frac{dy}{dt} =1$

I found some ways to solve equations like this with systems of equations in matlab, but I know nothing else about this system Can anyone offer me any insight or a push in the right direction?

I can seemingly find that $h(0) = ( \frac{b}{a} x(0))^2$ (given that dh/dt at time 0 is 0), but I'm not sure how this would be useful.

 

[mighty2000]

Thank you for sharing your problem with us. It seems that you are trying to solve for the maximum change in x(t) as y(t) changes within the given bounds. This type of problem can be solved using various methods, including ODE methods, MATLAB, or other numerical methods.

One approach you can try is to solve the equation numerically using MATLAB. You can use the built-in ODE solvers such as ode45 or ode23 to solve the equation and obtain the values of x(t) and y(t) for different time points. Once you have the values, you can plot them and visually determine the maximum change in x(t) as y(t) changes within the given bounds.

Another approach is to use the method of separation of variables. This involves rewriting the equation in the form of \frac{dy}{dt} = f(y)g(x), and then integrating both sides. This will give you an expression for x(t) in terms of y(t), which you can then use to calculate the maximum change in x(t) as y(t) changes.

I would also suggest considering the initial conditions of the problem, as they can provide valuable information in solving the equation. Additionally, you can try using a change of variables to simplify the equation and make it easier to solve.

I hope these suggestions will help you in approaching the problem. Good luck!