(Solved): Exercise 1. One aims to determine the time of death of a homicide victim. The body was found with a ...
Exercise 1. One aims to determine the time of death of a homicide victim. The body was found with a temperature of
85\deg Fat 12 h 00 PM in a closed room, which has a constant temperature of
72\deg F. Three hours later, the temperature of the body was measured in the same room and it was
78\deg F. Using Fourier's law of cooling, i.e.
T^(')(t)=k(T_(env)-T(t))where
kis a positive constant, find the time of death knowing also that at this time the victim had a normal body temperature of
98.6\deg F. Exercise 2. Find the general solution of each of the following equations: (1)
16y^('')-8y^(')+y=0(2)
y^('')+4y^(')+5y=0(3)
y^('')+4y^(')-5y=0(4)
y^('')-6y^(')+9y=(x+1)e^(3x)(5)
y^(''')-6y^('')+13y^(')-10y=e^(2x)cos(x)Exercise 3. Consider a linear homogeneous ODE of the form
T[y]=0, where the differential operator
Tis given as
T[y]=y^('')+a(x)y^(')+b(x)yfor some functions
a(x),b(x). (1) Prove that if
y_(1)(x)and
y_(2)(x)are two solutions of the given homogeneous equation, then so is every linear combination
c_(1)y_(1)(x)+c_(2)y_(2)(x)of
y_(1),y_(2)by scalars
c_(1),c_(2)inR. (2) Let
f(x)be another function. Consider now a non-homogeneous equation
T[y]=f(x)Suppose that
y_(p)(x)is a particular solution of the new equation. Prove that every other solution of
T[y]=f(x)is of the form
y=y_(p)+y_(h), where
y_(h)is a solution of the associated homogeneous equation.
