# Laplace Transform of Given Differential Equation

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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,

### Answers (2)

Sulaymon Eshkabilov
on 25 Apr 2022

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)

Sulaymon Eshkabilov
on 25 Apr 2022

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 = ilaplace(Sol,s,t)

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])

##### 1 Comment

Walter Roberson
on 25 Apr 2022

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