MATLAB: Finding roots of differential equations

differential equationslaplace transformsrootstransfer function

How can I find the roots/zeros of a differential equation such as

Best Answer

If I understand your Question correctly, you have to transform your equation from the time domain into Laplace space (assume all initial conditions are 0), then solve for the roots in s-space. Any book on signal processing or differential equations will tell you how to do the transformation, and will have tables of Laplace transforms.

The transfer function, H(s), will then be given by:

H(s) = U(s)/Y(s)

The ‘roots’ of that function will be the zeros of the numerator. In this second-order system, this becomes a straightforward application of the quadratic formula. If you do not have numerical values for ‘B’ and ‘C’, you will necessarily get only a symbolic result.

Related Solutions

MATLAB: How to showe laplace answer

Try this:

syms s t Y(s) y(t) Dy(t) 
DE=diff(y(t),t,2)+4*diff(y(t),t)+3*y(t);
LL=laplace(DE);
LL = subs(LL, {laplace(y(t), t, s), subs(diff(y(t), t), t, 0)},{Y(s),Dy(0)})

producing:

LL =
3*Y(s) - 4*y(0) - Dy(0) - s*y(0) + 4*s*Y(s) + s^2*Y(s)

MATLAB: Solving equations using Laplace transform

Unless you are solving a partial differential equation, such that the Laplace transform produces an ordinary differential equation in one of the two variables and a Laplace transform of ‘t’, dsolve is not appropriate. It is simply necessary to solve for (in this instance) ‘Y(s)’ and then invert it to get ‘y(t)’:

syms t y(t) Y(s) Dy0 
 
Dy2 = diff(y,t,2); 
Dy1 = diff(y,t,1); 
eqn = laplace(Dy2+3*Dy1+2*y == exp(-t))
eqn = subs(eqn, {laplace(y(t), t, s), subs(diff(y(t), t), t, 0)},{Y(s), Dy0})
cond = [y(0)==4, Dy1(0)==5]; 
Soln_s = isolate(eqn, Y(s))
Soln_t = ilaplace(Soln_s)
Soln_t = subs(Soln_t, {ilaplace(Y(s), s, t)}, {y(t)})
Soln_t = simplify(Soln_t, 'Steps',500)

Best Answer

Related Solutions

MATLAB: How to showe laplace answer

MATLAB: Solving equations using Laplace transform

Related Question