An implicit representation of a physiological system (i.e. in the form of a differential or\ difference equation) is often easier to derive from first principles than an explicit representation of\ the system (i.e. a function with output depending on an input). On the other hand, when analyzing\ the dynamics of a system, we often wish to solve the differential or difference equation and obtain\ its explicit representation.\ a. Many biopolymers used in biomedical engineering contain atoms strung together with\ covalent bonds. These bonds are known to vibrate with frequencies in the

10-100GHz

range\ and can be modeled with the following differential equation. Solve the following differential\ equation reflecting the spatial dynamics of two hydrogen atoms where

t

is in nanoseconds and\

r

is the distance between the two atoms:\

(d^(2)r(t))/(dt^(2))=-7\\\\times 10^(11)(r), given r(0)=7\\\\times 10^(-11) meters and (dr(0))/(dt)=0 meters / second

\ b. Digital filters, which we will talk more about in the coming weeks and months, are in\ many wearable sensors from heartrate monitors to inertial measurement unit devices. Consider\ the difference equation to filter an input electrocardiogram (ECG) signal,

x[n]

. Show the explicit\ (i.e. non-recursive) representation of the output

y[n]

in terms of only the input

x[n]

:\

y[n]-0.8y[n-1]+0.15y[n-2]=x[n] where x[n]=sin(10\\\\pi n)*u[n],y[0]=0, and y[1]=1

(24 pts) An implicit representation of a physiological system (i.e. in the form of a differential or difference equation) is often easier to derive from first principles than an explicit representation of the system (i.e. a function with output depending on an input). On the other hand, when analyzing the dynamics of a system, we often wish to solve the differential or difference equation and obtain its explicit representation. a. (12 pts) Many biopolymers used in biomedical engineering contain atoms strung together with covalent bonds. These bonds are known to vibrate with frequencies in the 10100GHz range and can be modeled with the following differential equation. Solve the following differential equation reflecting the spatial dynamics of two hydrogen atoms where t is in nanoseconds and r is the distance between the two atoms: dt2d2r(t)=71011(r),givenr(0)=71011metersanddtdr(0)=0meters/second b. (12 pts) Digital filters, which we will talk more about in the coming weeks and months, are in many wearable sensors from heartrate monitors to inertial measurement unit devices. Consider the difference equation to filter an input electrocardiogram (ECG) signal, x[n]. Show the explicit (i.e. non-recursive) representation of the output y[n] in terms of only the input x[n] : y[n]0.8y[n1]+0.15y[n2]=x[n] where x[n]=sin(10n)u[n],y[0]=0, and y[1]=1