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GATE 2016 EE - SET 2 - Question 4

\frac{\mathrm{d}y\left ( t \right ) }{\mathrm{d} t}+\frac{1}{6}y\left ( t \right )=3x\left ( t \right )

Applying Laplace transforms on both sides, L\left \{ \frac{\mathrm{d} y(t)}{\mathrm{d} t}+\frac{1}{6}y(t) \right \}=L\left \{ 3x(t) \right \}

\left [ s+\frac{1}{6} \right ]Y(S)=3X(S)

\frac{Y(S)}{X(S)}=\frac{3}{s+\frac{1}{6}}

\frac{Y(S)}{\left [ \frac{3}{S+\frac{1}{3}} \right ]}=\frac{3}{S+\frac{1}{6}}

Y(S)=\frac{3}{S+\frac{1}{6}}\ast \frac{3}{S+\frac{1}{3}} _______ (1)

Taking partial fractions, we get

Y(S)=\frac{A}{S+\frac{1}{6}}+\frac{B}{S+\frac{1}{3}} ________ (2)

And solving equations (1) and (2), we get the values of A and B, as, A = 54 and B = -54

So, equation (2) becomes, 

Y(S)=\frac{54}{S+\frac{1}{6}}+\frac{-54}{S+\frac{1}{3}}

Applying inverse laplace to the above equation, we get, 

y(t)=54e^{\frac{-t}{6}}u(t)-54e^{\frac{-t}{3}}u(t)

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