Solve the differential equation and find all the real solutions of the differential equation.
The solution is .
Consider the linear differential operator
The characteristic polynomial of 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.
Simplify further as follows.
Therefore, the roots of the characteristic polynomials are -1 and -1 .
Since, the roots of the characteristic equation is different real 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 T(f) = 0 .
Thus, the general solution of the differential equation 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.
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