Laplace Transform of Given Differential Equation

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Jordan Stanley 2022-4-25

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https://ww2.mathworks.cn/matlabcentral/answers/1704205-laplace-transform-of-given-differential-equation

移动：Sam Chak 2024-3-3

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Hello, I have the differential equation with initial condtions: y'' + 2y' + y = 0, y(-1) = 0, y'(0) = 0.

I need to use MATLAB to find the need Laplace transforms and inverse Laplace transforms.

I'm not sure if what I have so far is correct, here is what I have...

syms s t Y;
f = 0;
F = laplace(f,t,s);
Y1 = s*Y - 0;
Y2 = s*Y1 - 0;
laplaceSol = solve(Y2 + 2*Y1 + Y - F, Y)   %Laplace Transform
invlaplaceSol = ilaplace(laplaceSol,s,t)   %Inverse Laplace Transform

I get the following as output.

laplaceSol = 0
invlaplaceSol = 0 

I also have the following code in an m-file.

function myplot(f,interv)
% myplot(f,[a,b])
% plot f for interval [a,b]
% here f is a symbolic expression, or a string
%
% example:
%  myplot('x^2',[-1,1])
% syms x; myplot(x^2,[-1,1])
f = sym(f);
tv = linspace(interv(1),interv(2),300);
T = findsym(f,1);
plot(tv,double(subs(f,T,tv)))

Thank you,

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MD.AL-AMIN 2024-3-2

移动：Sam Chak 2024-3-3

y''(t) + 4y'(t) + 8y(t) = x' (t)+x(t) with x(t) = e ^ (- 4t) * u(t) y(0) = 0 and y'(0) = 0 matlab code

MD.AL-AMIN 2024-3-2

移动：Sam Chak 2024-3-3

I don't know how to solve by using MATLAB?

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回答（2 个）

Sulaymon Eshkabilov 2022-4-25

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https://ww2.mathworks.cn/matlabcentral/answers/1704205-laplace-transform-of-given-differential-equation#answer_950155

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Laplace transform does not work at t ~0 initial conditions and thus, here dsolve() might be a better option, e.g.:

syms y(t)

Dy=diff(y,t);

D2y = diff(y,t,2);

Eqn = D2y == -2*Dy-y;

ICs = [y(-1)==0, Dy(0)==0];

S = dsolve(Eqn, ICs)

S =

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Walter Roberson 2022-4-25

Though, considering that dsolve() using Dy(0) == 0 gives the same solution as dsolve() with y(-1)==0, then you could potentially ignore the y(-1) and go ahead with laplace.

Jordan Stanley 2022-4-25

I'm not sure how to include the initial conditions when using the laplace() function.

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Sulaymon Eshkabilov 2022-4-25

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1704205-laplace-transform-of-given-differential-equation#answer_950825

在 MATLAB Online 中打开

Note that if your system has "zero" ICs and not excitation force; therefore, your system solution (response) will be zero. If you set one of your ICs, non-zero varlue and then you'll see something, e.g.:

syms s Y t

y0=0;

dy0=-1; %

Y1 = s*Y - y0;

Y2 = s*Y1- dy0;

Sol = solve(Y2 + 2*Y1 + Y, Y)

Sol =

Sol = ilaplace(Sol,s,t)

Sol =

fplot(Sol, [-1, 1])

% Verify: alternative solution with dsolve() gives the same result

syms y(t)

Dy=diff(y,t);

D2y = diff(y,t,2);

Eqn = D2y == -2*Dy-y;

ICs = [y(0)==0, Dy(0)==-1];

S = dsolve(Eqn, ICs);

fplot(S, [-1, 1])

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Walter Roberson 2022-4-25

However... the posters have been clear that they have an initial condition at y(-1) not an initial condition at y(0) . Which is a problem for laplace transforms.

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