(Solved): Problem 2: Consider the LTI systems described by the following differential equations. A) Compute t ...

Problem 2: Consider the LTI systems described by the following differential equations. A) Compute the impulse response and the unit-step response of each system using the Fourier transform: 1) $y^{?}(t)+7y(t)=x(t),y(0)=0$ - Find the output $y(t)$ corresponding to the input $x(t)=e^{?t}t?0$. 2) $y^{??}(t)+7y^{?}(t)+12y(t)=x(t),y(0)=y^{?}(0)=0$ - Find the output $y(t)$ corresponding to the input $x(t)=e^{?t},t?0$. 3) $y^{??}(t)+7y^{?}(t)+12y(t)=x^{?}(t),y(0)=y^{?}(0)=0$ - Find the output $y(t)$ corresponding to the input $x(t)=e^{?t},t?0$. 4) $y^{??}(t)+7y^{?}(t)+12y(t)=x^{?}(t)+x(t),y(0)=y^{?}(0)=0$ 5) $y^{??}(t)+7y^{?}(t)+12y(t)=x^{??}(t),y(0)=y^{?}(0)=0$ 6) $y^{??}(t)+12y(t)=x(t),y(0)=y^{?}(0)=0$. 7) $y^{??}(t)?7y^{?}(t)+12y(t)=x(t),y(0)=y^{?}(0)=0$. 8) $y^{??}(t)+7y^{?}(t)?12y(t)=x(t),y(0)=y^{?}(0)=0$. B) Which systems are stable?