How to Solve an ODE Problem when one of parameters is dependent to derivative?

In summary, you can solve an equation in form of dy/dt=f(t,y,dy/dt) by using an algorithm of this kind: Start with given initial values y and t. Recursive process : compute A(y,t), B(y,t), C(y,t) and D(y,t) . Solve X=A+B+f(C,D,X) with a numerical equation solver. With the computed value X=dy/dt the incrementation of y is done, as well as the incrementation of t.
  • #1
mahdi_zabchek
4
0
Hello Guys!
I have an ODE problem that I'm solving it by MATLAB ODE solvers!
in fact I have a system of non-linear differential equations in one of these equations I have a parameter that it's value is dependent to derivative! the general form of equation is like this (big letter parameters are known!):

dy/dt = A + B + f(C,D,dy/dt)

how can I solve this problem by ode45 or other MATLAB ODE solvers?
 
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  • #2
Is the function f known?
 
  • #3
hunt_mat said:
Is the function f known?

yes! it is.
but it's not reversible
 
  • #4
mahdi_zabchek said:
Hello Guys!
I have an ODE problem that I'm solving it by MATLAB ODE solvers!
in fact I have a system of non-linear differential equations in one of these equations I have a parameter that it's value is dependent to derivative! the general form of equation is like this (big letter parameters are known!):

dy/dt = A + B + f(C,D,dy/dt)

how can I solve this problem by ode45 or other MATLAB ODE solvers?

The ODE : dy/dt = A + B + f(C,D,dy/dt) contains no y and no t. As a consequence dy/dt = constant.
Let X= dy/dt . X is solution of the equation X = A + B + f(C, D, X) which is not an ODE.
It doesn't matter if the function is not revertsible. We don't need to know the analytical expression of the solution(s) X. We know that dy/dt = constant (or = several different constants if there are several solutions). Each one can be numerically computed, not using an ODE solver, but using an usual numerical equation solver.
The solution(s) is (are) : y(t) = X*t +c
c is a constant to be determined by the boundary condition.
 
  • #5
JJacquelin said:
The ODE : dy/dt = A + B + f(C,D,dy/dt) contains no y and no t. As a consequence dy/dt = constant.
Let X= dy/dt . X is solution of the equation X = A + B + f(C, D, X) which is not an ODE.
It doesn't matter if the function is not revertsible. We don't need to know the analytical expression of the solution(s) X. We know that dy/dt = constant (or = several different constants if there are several solutions). Each one can be numerically computed, not using an ODE solver, but using an usual numerical equation solver.
The solution(s) is (are) : y(t) = X*t +c
c is a constant to be determined by the boundary condition.

No! No! it has y and t!
A and B and C and D are NOT constant parameters!
I did't write them because they were not necessary!
in fact You don't need to know what's the equation exactly to answer my question!

My question is simple:

MATLAB ODE solvers solve equations in form of dy/dt=f(t,y) but I want to solve an equation in form of dy/dt=f(t,y,dy/dt) ... How I can do that by MATLAB?
 
  • #6
mahdi_zabchek said:
No! No! it has y and t!
A and B and C and D are NOT constant parameters!
I did't write them because they were not necessary!
in fact You don't need to know what's the equation exactly to answer my question!

My question is simple:

MATLAB ODE solvers solve equations in form of dy/dt=f(t,y) but I want to solve an equation in form of dy/dt=f(t,y,dy/dt) ... How I can do that by MATLAB?

OK. Sorry for the missunderstanding.
May be, you could use an algorithm of this kind:
Start with given initial values y and t.
Recursive process :
Compute A(y,t), B(y,t), C(y,t) and D(y,t)
Solve X=A+B+f(C,D,X) with a numerical equation solver, introduced as sub-program.
With the computed value X=dy/dt the incrementation of y is done, as well as the incrementation of t.
Then continue the recursive process.
 
  • #7
JJacquelin said:
OK. Sorry for the missunderstanding.
May be, you could use an algorithm of this kind:
Start with given initial values y and t.
Recursive process :
Compute A(y,t), B(y,t), C(y,t) and D(y,t)
Solve X=A+B+f(C,D,X) with a numerical equation solver, introduced as sub-program.
With the computed value X=dy/dt the incrementation of y is done, as well as the incrementation of t.
Then continue the recursive process.

I'll try it ... thank you so much
 

1. How do I solve an ODE problem when one of the parameters is dependent on the derivative?

Solving an ODE problem when a parameter is dependent on the derivative requires using the method of separation of variables. This means separating the equation into two parts, one with the dependent parameter and its derivative, and the other with the independent variables. Then, you can integrate both parts and solve for the dependent parameter.

2. Can I use substitution to solve an ODE problem with a dependent parameter?

Yes, substitution can be used to solve an ODE problem with a dependent parameter. By substituting the dependent parameter with a new variable, you can solve the resulting equation as you would a regular ODE problem.

3. Are there any specific techniques for solving ODE problems with dependent parameters?

Yes, there are several techniques that can be used to solve ODE problems with dependent parameters, such as the method of integrating factors, the variation of parameters method, and the method of undetermined coefficients. It is important to choose the appropriate technique based on the specific problem at hand.

4. Is it possible to have multiple dependent parameters in an ODE problem?

Yes, it is possible to have multiple dependent parameters in an ODE problem. In this case, you will need to use a combination of the aforementioned techniques, such as separation of variables and substitution, to solve the equation.

5. How do I know if my solution for an ODE problem with a dependent parameter is correct?

To ensure the correctness of your solution, you can plug it back into the original ODE equation and check if it satisfies the equation. Additionally, you can also compare your solution to other known solutions or use a computer program to verify your answer.

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