How Can I Solve a Unique First-Order ODE Numerically?

[makris]

I would like to solve a problem of the type

(da/dt)^2 + f(a)* (da/dt) = g(a) (1)

a=a(t) unknown function
f(a), g(a) = known functions of a.

This differential equation is a first order ODE but (da/dt)^2 makes it different compared to a typical first order ODEs (at least to my knowledge)

I would like to find a(t) satisfying (1) subject to certain initial conditions (say a(0.1)=2).

I feel that no appropriate analytical solution exists for this type of problem, so I am looking for a numerical method to integrate it.

I am thinking of setting da/dt=y thus having
---------------------------
y^2 + f(a)* y = g(a)
da/dt=y
----------------------------

and then writing da/dt = ( a(i+1) - a(i) ) /dt

so the problem becomes
---------------------------
y^2 + f(a(i))* y = g(a(i)) (2)
a(i+1) = dt*y + a(i) (3)
----------------------------

Now I am thinking of solving (2) for the value of y which corresponds at i=0 and then keep one the two solutions (which one to keep is not very clear ….(or if they are both imaginaray?)) Then with a selected small dt (say dt=0.001) find a(i+1). Then continue the iteration scheme this way.

I know a priori that da/dt is positive and thus a(t) is an increasing function of t.

I would like to have opinion from you whether the previous reasoning is TOTALLY WRONG or not. If it wrong I would appreciate if you just give a hint of how to attack the problem

Thanks

 

[Tide]

HINT: Use the quadratic formula. :)

 

[tacman]

Your reasoning is not totally wrong, but there are a few things to consider in solving this type of problem. First, the equation (1) is a second order ODE, not a first order ODE. This is because it contains the second derivative of a.

To solve this problem numerically, you can use a method called the fourth-order Runge-Kutta method. This method is commonly used for solving second order ODEs and can also be used for higher order equations.

To use this method, you will need to rewrite (1) as a system of two first order equations. This can be done by setting y = da/dt, as you have already done. Then, you can rewrite (1) as:

dy/dt = -f(a)*y + g(a)
da/dt = y

Now, you can use the fourth-order Runge-Kutta method to solve this system of equations. This method involves calculating four intermediate values of y and a at each step, using the previous values to calculate the next values. The steps for this method are as follows:

1. Start with initial values for y and a, given by the initial conditions (a(0.1) = 2 in this case).
2. Calculate the intermediate values of y and a at time t = 0.1 using the above equations.
3. Use these intermediate values to calculate the next values of y and a at time t = 0.2.
4. Repeat this process until you reach the desired time.
5. Plot the values of a and y as a function of time to see the solution.

It is important to note that for this method to work, the functions f(a) and g(a) must be continuous and smooth. Also, the step size dt you choose will affect the accuracy of the solution. You may need to adjust the step size to get a more accurate solution.

In summary, your reasoning is on the right track, but you will need to use a different method to solve this problem. The fourth-order Runge-Kutta method is a good choice for solving second order ODEs numerically. I hope this helps!