Comment utiliser les transformées de Laplace ? How to use Laplace transforms

Résolution d'équations différentielles par la transformée de Laplace avces Matlab . Solving differential equations using Laplace transforms. ????

Answer 1

[CHANDRA BABU GUTTIKONDA]

Solving differential equations using the Laplace transform in MATLAB involves converting a time-domain differential equation into an algebraic equation in the s-domain, which simplifies the solution process, especially for linear systems with given initial conditions. The method applies the Laplace transform to each term of the differential equation, uses standard transform properties to incorporate initial conditions directly, solves the resulting algebraic equation for the transformed variable Y(s)Y(s)Y(s), and then applies the inverse Laplace transform to obtain the time-domain solution y(t)y(t)y(t). In MATLAB, this is efficiently implemented using symbolic functions such as laplace, ilaplace, diff, and solve, making the approach particularly suitable for analyzing electrical circuits, mechanical systems, and control systems where transient and steady-state responses are of interest.

CODE FOR SOLVING SECOND ORDER DIFFERNTIALEQUATION

syms y(t) s

Dy = diff(y,t);
D2y = diff(y,t,2);

eqn = D2y + 4*Dy + 5*y == 0;

Y = laplace(y,t,s);
eqn_s = laplace(eqn,t,s);

eqn_s = subs(eqn_s, laplace(y,t,s), Y);
eqn_s = subs(eqn_s, laplace(Dy,t,s), s*Y - 1);
eqn_s = subs(eqn_s, laplace(D2y,t,s), s^2*Y - s*1 - 0)

eqn_s = 

Y_sol = solve(eqn_s, Y);

Warning: Unable to find explicit solution. For options, see help.

y_sol = ilaplace(Y_sol, s, t);
simplify(y_sol)

 
ans =
 
Empty sym: 0-by-1
 

[CHANDRA BABU GUTTIKONDA]

in Simulink, this is implemented by placing a Transfer Fcn block with numerator [1] and denominator [1 3], feeding it with a Step input of amplitude 6, and observing the output on a Scope block. Internally, Simulink converts the Laplace model into state-space form for numerical simulation, making Laplace blocks especially useful for control systems, signal processing, and power electronics modeling where system dynamics are naturally expressed using transfer functions.

[Paul]

Perhaps this answer should be corrected so that it doesn't return ysol as an empty result and show an empty plot.

Answer 2

[Paul]

Hi Gilles,

The documentation provides a detailed example at Solve Differential Equations of RLC Circuit Using Laplace Transform. That example is for two, first order differential equations, but it should be adaptable to other types of problems, like one, higher order differential equation.

Also, you can find several Questions about this exact problem by searching here on Answers.

If you still have questions, feel free to post back and show the equation(s) you're trying to solve with the code you're developing and explain/illustrate what problems you are having.