(Solved): n the series LR circuit given above the current \( i(t) \) is produced by the driving voltage \( v( ...

n the series LR circuit given above the current \( i(t) \) is produced by the driving voltage \( v(t) \). i) Derive the differential equation which must be solved in order to obtain an expression \( i(t) \) for the current as a function of \( t \) for any given input voltage \( v(t) \). ii) Derive, using the brute force method of directly solving the foregoing differential equation, an expression for the impulse response for the current is(t) \( =\mathrm{h}(\mathrm{t}) \) which results when \( v(t)=8(t) \), with \( h(t) \) being the conventional symbol used for the impulse response of an LTI system or circuit. Assume the initial current \( i\left(0^{-}\right) \)to be zero. iii) Now assume that the input voltage is \( v(t) \) is the unit step voltage \( v(t)=u(t) \). Derive an expression for the current ius(t) which is obtained when the unit step voltage \( v_{i}(t)=u(t) \) drives the circuit. Assume the initial current at \( t=0 \) is zero. You are to derive the expression for ius(t) by three methods: Method A: Brute force method of directly solving the differential equation for \( i(t) \). Method B: Directly integrating the impulse response h(t). Show that this method is valid by integrating the differential equation in ii) so that the driving voltage on the right hand side of the differential equation converts from \( \delta(t) \) to \( u(t) \). Methode \( \mathrm{C} \) : By using the convolution method which requires finding the convolution of two signals one of which is the impulse response signal h(t) - i.(t) and the other is the new unit step input voltage signal u(t).