Solution of ODE with Symbolic Math Toolbox.

Hello,

I want to solve the ordinary differential equation

with , , . The function and should be a unity step. In other words: it is the step response of a first order lag element. I am using the Symbolic Math Toolbox.

First I define the variables including the relevant assumptions.

syms("t", "positive")
syms("T_1", "k_p", "positive")
syms("u(t)", "y(t)")

The I establish the differential equation

deq = T_1 * diff(y, t) + y == k_p * u

deq(t) = 

and replace with the unist step function.

deq = subs(deq, u, heaviside(t))

deq(t) = 

The solution

y_sol_t = dsolve(deq, y(0) == 0)

y_sol_t = 

y_sol_t = simplify(y_sol_t)

y_sol_t = 

is not very clear for students. It is hard to see that for .

If I use the Laplace Transform instead I get: the following result.

Laplace Transform of the above differential equation.

syms("s")
aeq = laplace(deq, t, s)

aeq = 

syms("y_LT")

Replacing laplace(y(t), t, s) and the initial condition

aeq = subs(aeq, [laplace(y(t), t, s) y(0)], [y_LT, 0])

aeq = 

Solve the algbraic equation vor y_LT

y_LT = solve(aeq, y_LT)

y_LT = 

Inverse Laplace Transform

y_sol_LT = ilaplace(y_LT, s, t)

y_sol_LT = 

which is valid only for , but can be extended to the general solt ion by multiplying it with ethe unit step.

y_sol_LT = y_sol_LT * heaviside(t)

y_sol_LT = 

How can I obtain this solution from the result calculated using dsolve?

Michael

Answer 1

[Paul]

Use rewrite to get the solution in terms of heaviside.

syms("t", "positive")
syms("T_1", "k_p", "positive")
syms("u(t)", "y(t)")
deq = T_1 * diff(y, t) + y == k_p * u;
deq = subs(deq, u, heaviside(t));
y_sol_t = dsolve(deq, y(0) == 0)

y_sol_t = 

y_sol_t = simplify(y_sol_t)

y_sol_t = 

y_sol_t = rewrite(y_sol_t,'heaviside')  % makes clear the solution is 0 for t < 0

y_sol_t = 

To get the exact same result as from Laplace ... an easy, if not elegant, approach

simplify(y_sol_t/heaviside(t))*heaviside(t)

ans = 

[Michael]

Thank you for showing mne the use of rewrite with the heaviside argument. Altough I am using this function in aother contexts, I was not aware of using it with heaviside.

After ereading your approach; I was playing alittle bit with collect and combine statements, however your easy approach in the last line is the only one giving the expected result.

What I simply don't understand: I make an assumption for the variable t, which should be poitive and I set an initial condition for t=0. So it should be obvious, that the interesting solution should start at t=0.

Michael