Rename integration variable in symbolic result

Hi,

There is an ambiguity in the name of vaiable x.

In the code below, I use x is a function of the variable t and Matlab uses it as the variable of integration.

How to change the name of the variable of integration ?

syms t m b y(t) y0 C x(t) % definitions of symbolic variables 
cond = y(0) == C  % inital condition 
ode = m*diff(y,t) + b*y(t) == x(t)   % definition of differential equation
ySymbSol(t) = dsolve(ode, cond)  % Solution of the differential equation which uses x as integration variable !

The same result is obtained with Matlab 2022a and 2022b.

[Torsten]

Should be easier to rename your function x, shouldn't it ?

[Mohamed Gharbi]

Yes. but in the book I'm writing, I use as notation for linear systems x(t) for input and y(t) for output. I would like, if possible, to keep these notations.

[Torsten]

I doubt you can specify it easily.

Answer 1

[Paul]

I don't know if I'd call this 'easily', but it gets the desired result (I believe).

syms t m b y(t) y0 C x(t) % definitions of symbolic variables 
cond = y(0) == C;  % inital condition 
ode = m*diff(y,t) + b*y(t) == x(t);   % definition of differential equation
syms u(t)
ySymbSol(t) = dsolve(subs(ode,x(t),u(t)), cond)  % Solution of the differential equation which uses x as integration variable !

ySymbSol(t) = 

syms syms x tau
ySymbSol(t) = changeIntegrationVariable(ySymbSol(t),x,tau)

ySymbSol(t) = 

syms u
ySymbSol(t) = subs(ySymbSol(t),u,x)

ySymbSol(t) = 

syms x(t)

[Torsten]

We say: From behind through the chest into the eye :-)

Answer 2

[Surya]

Hi,

To avoid the ambiguity in the variable name, you can use a different variable for integration in the dsolve command. You can do this by providing the integration variable as the second argument to the dsolve function. Here's an example:

syms t m b y(t) y0 C x(t) tau
cond = y(0) == C; % initial condition
ode = m*diff(y,t) + b*y(t) == x(t); % definition of differential equation
ySymbSol(tau) = dsolve(ode, cond, tau); % solve the ODE with respect to tau

In this example, the dsolve function is called with the third argument set to tau, which specifies that tau is the independent variable in the differential equation. The second argument t is used as the integration variable in the solution. The resulting ySymbSol function will depend on tau instead of t.

[Torsten]

This wasn't the problem. It was the x as formal integration variable which conflicts with the function x(t).

syms t m b y(t) y0 C x(t) tau
cond = y(0) == C; % initial condition
ode = m*diff(y,t) + b*y(t) == x(t); % definition of differential equation
ySymbSol(tau) = dsolve(ode, cond, tau) % solve the ODE with respect to tau

ySymbSol(tau) = 

[Surya]

Here x(t) is renamed as g(t)

syms t m b y(t) y0 C g(t) % definitions of symbolic variables 
cond = y(0) == C  % inital condition 
ode = m*diff(y) + b*y(t) == g(t)   % definition of differential equation
ySymbSol(t) = dsolve(ode, cond)

And here is the final function

Now x becomes the variable of integration and g(x) is the function of x taking values from [ 0, t ].

Hope it helps.

[Torsten]

@Surya

See my discussion with the OP after the original question:

My comment was:

Should be easier to rename your function x, shouldn't it ?

And his answer was:

Yes. but in the book I'm writing, I use as notation for linear systems x(t) for input and y(t) for output. I would like, if possible, to keep these notations.

And my comment was:

I doubt you can specify it easily.

Or is it possible to specify it easily ?