Question

A differential equation is an equation which involves an unknown function along with its derivatives (this will be covered in more depth in MATH 204). For example, dy/dx = y^2 where y is as unknown function is a differential equation. A function that satisfies the above equation if substitution on both sides of the equation yields an identity is called a solution to the differential equation. For example, substituting the function y = -1/x in both sides of the differential equation yields: dy/dx = y^2 d/dx(-1/x) = (-1/x)^2 1/x^2 = 1/x^2 thus, y = -1/x is a solution to the differential equation dy/dx = y^2. In Biology, population dynamics can be modeled with differential equations. The exponential growth model is given by the differential equation dN/dt = rN, N(0) = N_0 where r is the per-capita growth rate of the population, N(t) is the population size at a given time t, and N(0) = N_0 is the initial population. This equation says that the rate at which the population grows is proportional to its current population (the larger the population size is at a given time, the more the population can reproduce). Show that the function given by N(t) = N_0e^rt is a solution to the differential equation. (Hint: Compute the derivative of N(t) treating N_0 and r as constants.)

          A differential equation is an equation which involves an unknown function along with its derivatives (this will be covered in more depth in MATH 204). For example,

dy/dx = y^2

where y is as unknown function is a differential equation. A function that satisfies the above equation if substitution on both sides of the equation yields an identity is called a solution to the differential equation. For example, substituting the function y = -1/x in both sides of the differential equation yields:

dy/dx = y^2
d/dx(-1/x) = (-1/x)^2
1/x^2 = 1/x^2

thus, y = -1/x is a solution to the differential equation dy/dx = y^2.

In Biology, population dynamics can be modeled with differential equations. The exponential growth model is given by the differential equation

dN/dt = rN, N(0) = N_0

where r is the per-capita growth rate of the population, N(t) is the population size at a given time t, and N(0) = N_0 is the initial population. This equation says that the rate at which the population grows is proportional to its current population (the larger the population size is at a given time, the more the population can reproduce).

Show that the function given by

N(t) = N_0e^rt

is a solution to the differential equation. (Hint: Compute the derivative of N(t) treating N_0 and r as constants.)

A differential equation is an equation which involves an unknown function along with its derivatives (this will be covered in more depth in MATH 204). For example,

dy/dx = y^2

where y is as unknown function is a differential equation. A function that satisfies the above equation if substitution on both sides of the equation yields an identity is called a solution to the differential equation. For example, substituting the function y = -1/x in both sides of the differential equation yields:

dy/dx = y^2
d/dx(-1/x) = (-1/x)^2
1/x^2 = 1/x^2

thus, y = -1/x is a solution to the differential equation dy/dx = y^2.

In Biology, population dynamics can be modeled with differential equations. The exponential growth model is given by the differential equation

dN/dt = rN, N(0) = N0

where r is the per-capita growth rate of the population, N(t) is the population size at a given time t, and N(0) = N0 is the initial population. This equation says that the rate at which the population grows is proportional to its current population (the larger the population size is at a given time, the more the population can reproduce).

Show that the function given by

N(t) = N0e^rt

is a solution to the differential equation. (Hint: Compute the derivative of N(t) treating N0 and r as constants.)

Added by Chelsey R.

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