Laplace transform of sawtooth function for 2nd order ode

Hi all,

I have taken the rhs laplace for a sawtooth equation where:

f(t)=2t for 0<t<1

and f(t+1)=f(t)

T=2

>> syms s t lapf
lapf =simplify(int('exp(-s*t)*2*t','t=0 .. 2')/(1-exp(-s)))
pretty(lapf)
 
lapf =
 
-(2/s^2 - (2*(2*s + 1))/(s^2*exp(2*s)))/(1/exp(s) - 1)

Now I need to solve the differential equation: y'' + y = f(t)

Here, f(t) is the sawtooth function above.

I am having some difficulty marrying the two. Any suggestions would be appreciated.

Thank you!

[Aquatris]

You take the laplace transform of both sides of the equation (assuming 0 initial conditions);

s^2*Y(s)+Y(s) = F(s)
Y(s) = F(s)/(s^2+1)

Then do the algebra and apply inverse laplace to find the solution in time domain. If there are nonzero initial conditions, use them in the laplace transform.

[spcrooks]

Thank you for the response, Aquatris.

My trouble isn't with the analytical approach to the equation, but rather how to properly code the lhs and rhs of the equation together. Sorry, I realize that my initial post may not have been clear in that regard.

[Aquatris]

I don't understand your question. Do you want to numerically solve it? In that case you can define the equation as a state space equation if you are familiar with it;

% Time domain equation; y''+y = f 
% Select system states; x = [y y']'
% State Space equation; x' = Ax + Bu;
%                       y = Cx
A = [0 1;-1 0];
B = [0 1]';
C = eye(2); 
sys = ss(A,B,C,0);
t = 0:0.001:10; % time vector;
u = sawtooth(t);% input sawtooth vector
y = lsim(sys,u,t);

[spcrooks]

I thought about your initial post earlier, and that led me to this...

>> syms s t Y
f = f(t);
F = laplace(f,t,s);
Y1 = Y;
Y2 = Y*s^2;
sol = solve(Y2 + Y1 - F, Y);

 

However, now I need to figure out how to appropriately insert f(t), which sawtooth function from the OP. I have tried the code with a dummy value for "f" and it works.

[Aquatris]

If thats what you want to do, and you are sure about the laplace transform of your sawtooth function, then the answer is easy;

Y = -(2/s^2 - (2*(2*s + 1))/(s^2*exp(2*s)))/(1/exp(s) - 1)/(s^2+1)

You already determined F in your question (the variable lapf), so why are you confused?