Solve the initial value problem in
The solution is.
Consider the linear differential operator
The characteristic polynomial of T is defined as
The characteristic polynomial of the operator as follows.
Then the characteristic polynomial is as follows.
Solve the characteristic polynomial and find the roots as follows.
Therefore, the roots of the characteristic polynomials are and .
Since, the roots of the characteristic equation are different complex numbers.
The exponential functions and form a basis of the kernel of T .
Hence, they form a basis of the solution space of the homogenous differential equation is .
Thus, the general solution of the differential equation is .
Consider the general solution of the with the initial value problem is as follows,
Hence the solution is .
Question: Consider the interaction of two species in a habitat. We are told that the change of the populations can be moderated by the equation
where time is a measured in years.
Question: In 1778, a wealthy Pennsylvanian merchant named Jacob De Haven lent $450,000 to the continental congress to support the troops at valley Forge. The loan was never repaid. Mr De Haven’s descendants have taken the U.S. government to court to collect what they believe they are owed. The going interest rate at the time was 6%. How much were the De Havens owed in 1990
(a) if interest is compounded yearly?
(b) if interest is compound continuously?
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