# Transfer function with 2 inputs and 2 outputs

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meng fowler on 8 Dec 2020
Answered: Ameer Hamza on 8 Dec 2020
Hello,
My differential equation is x''(t)=(562/101)*x(t)+(6744/101)*I(t)+(1/50500)*F(t)
outputs ---->x , v(velocity)
inputs-----> I,F
The Transfer function is
g11=6744/(101s^2-562)
g12=1/(50500s^2-281000)
g21=6744*s/(101s^2-562)
g22=s/(50500s^2-281000)
I can find the steps to prove the transfer function for g11 and g12 using LaPlace, but i cant prove the s on the numerator of g21,g22.

Ameer Hamza on 8 Dec 2020
Study this code
syms x(t) v(t) I(t) F(t) s
v(t) = diff(x, t);
eq = diff(v) == (562/101)*x+(6744/101)*I+(1/50500)*F;
lap = laplace(eq);
S = subs(lap, [x(0) v(0)], [0 0]); % initial conditions should be zero
syms lx lv lI lF
Sx = subs(S, [laplace(x(t), t, s) laplace(I(t), t, s) laplace(F(t), t, s)], [lx lI lF]);
Sv = subs(S, [laplace(x(t), t, s) laplace(I(t), t, s) laplace(F(t), t, s)], [lv/s lI lF]);
g11 = simplify(lx/solve(subs(Sx,lF,0),lI))
g12 = simplify(lx/solve(subs(Sx,lI,0),lF))
g21 = simplify(lv/solve(subs(Sv,lF,0),lI))
g22 = simplify(lv/solve(subs(Sv,lI,0),lF))