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, 

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