How can I determine the order of a symbolic differential equation?

I'm writing a function that takes a differential equation in symbolic form as an argument and I want to determine the order of the equation in terms of certain variables.

Example 1 A first-order system:

syms t s y(t) u(t) R L
diff_eqn = R*y(t) + L*diff(y(t), t) == u(t);  % differential equation

The order (w.r.t. y(t)) is 1.

Example 2 A second-order system:

syms t s y(t) u(t) omega_n z K
diff_eqn = 1/omega_n^2*diff(y(t), t, 2) + 2*z/omega_n*diff(y(t), t) + y(t) == K*u(t);

The order is 2.

I would also like to know the order w.r.t. u(t) if possible as well which in general might not be 0.

Answer 1

[Ayush Gupta]

There doesn’t exist a direct function to determine the order of a differential equation. However, there is a workaround, and we can use the reduceDifferentialOrder and get newvars from where we can get the last element and see the occurrence of t and this is one plus than the order of equation. Refer to the following code:

syms x(t) y(t) f(t) 
eqs = [diff(x(t),t,t) == diff(f(t),t,t,t), diff(y(t),t,t,t) == diff(f(t),t,t)];  
vars = [x(t), y(t)]; 
[newEqs, newVars, R] = reduceDifferentialOrder(eqs, vars) 
l = length(newVars);
s = string(newVars(l,1));  
order_of_equation = count(s, 't') -1; 

[Paul]

It's a bit easier if dealing wth ODEs as in the Question, at least to find the order of the ODE

syms t s y(t) u(t) R L
diff_eqn = R*y(t) + L*diff(y(t), t) == u(t)  % differential equation

diff_eqn = 

%The order (w.r.t. y(t)) is 1.
numel(odeToVectorField(diff_eqn))

ans = 1

% Example 2 A second-order system:
syms t s y(t) u(t) omega_n z K
diff_eqn = 1/omega_n^2*diff(y(t), t, 2) + 2*z/omega_n*diff(y(t), t) + y(t) == K*u(t)

diff_eqn = 

numel(odeToVectorField(diff_eqn))

ans = 2

Finding the order wrt u(t) would take more work.

Answer 2

[Ganesh]

function order = order_polynomial(poly1,x)
count=0;
temp=1;
while(temp~=0)
    
    poly1=diff(poly1,x);
    if poly1==0
        temp=0;
    else
        count=count+1;
    end
end
disp(count);
end

[Ganesh]

this is for finding the order of polynomial function

[Walter Roberson]

You can use coeffs to get the information without a loop

Answer 3

[Walter Roberson]

p = poly2sym(randi([-9,9],1,randi(15)))

p = 

poly_degree = length(coeffs(p,'all'))-1

poly_degree = 6

[Bill Tubbs]

This answer is great, thanks. I think it should be written as a separate answer.

[Paul]

If the coefficient vector is double there's no need to go the Symbolic route.

rng('default')
c = [0 0 0 randi([-9,9],1,randi(15))]  % add some leading zeros

c = 1×16

0 0 0 8 -7 8 3 -8 -4 1 9 9 -7 9 9 0

tic
for ii = 1:1e3
    p = poly2sym(c);
    poly_degree_s = length(coeffs(p,'all'))-1;
end
toc

Elapsed time is 2.962911 seconds.

tic
for ii = 1:1e3
poly_degree_d = numel(c) - find(c,1,'first');
end
toc

Elapsed time is 0.004062 seconds.

[poly_degree_s poly_degree_d]

ans = 1×2

12 12

[Walter Roberson]

@Paul is of course correct that if you have a vector of coefficients then the difference between the length of the vector and the position of the first non-zero tells you about the degree.

However... the original question deals with symbolic polynomials. My creation of p with intermediate numeric form was just to have some polynomial to work with, and to demonstrate that I my code worked with polynomials of different degrees, not just something that "happened" to work with a particular length.