This question is asking a population dynamic type of scenario.

The total population in terms of time and where is the constant rate of proportionality, is described by the following differential equation.

For this particular function it's known that the population is in the form to represent the dogs. Since the virus spreads based on the interactions between the dogs who have the virus and those who have not yet contracted it, the population will be the product of the number of dogs with the virus and those without the virus.

Therefore the differential equation becomes,

This problem contains two questions that need to be solved for: order of the differential equation and whether it is linear or nonlinear.

To determine the order of the differential equation, look for the highest derivative in the equation.

For this particular function recall that,

therefore the highest derivative is three which makes the equation a third ordered differential equation.

The second part of this problem is to determine if the equation is linear or nonlinear. For a differential equation to be linear two characteristics must hold true:

1. The dependent variable and all its derivatives have a power involving one.

2. The coefficients depend on the independent variable .

Looking at the given function,

it is seen that all the variable and all its derivatives have a power involving one and all the coefficients depend on therefore, this differential equation is linear.

To answer this problem completely, the differential equation is a linear, third ordered equation.

So this is a separable differential equation. We can think of as

and as .

Taking the anti-derivative once, we get:

Then using the initial condition

We get that

So Then taking the antiderivative one more time, we get:

and using the initial condition

we get the final answer of:

This problem contains two questions that need to be solved for: order of the differential equation and whether it is linear or nonlinear.

To determine the order of the differential equation, look for the highest derivative in the equation.

For this particular function recall that,

therefore the highest derivative is three which makes the equation a third ordered differential equation.

The second part of this problem is to determine if the equation is linear or nonlinear. For a differential equation to be linear two characteristics must hold true:

1. The dependent variable and all its derivatives have a power involving one.

2. The coefficients depend on the independent variable .

Looking at the given function,

it is seen that all the variable and all its derivatives have a power involving one and all the coefficients depend on therefore, this differential equation is linear.

To answer this problem completely, the differential equation is a linear, third ordered equation.

So this is a separable differential equation. We can think of as

and as .

Taking the anti-derivative once, we get:

Then using the initial condition

We get that

So Then taking the antiderivative one more time, we get:

and using the initial condition

we get the final answer of:

This question is asking a population dynamic type of scenario.

The total population in terms of time and where is the constant rate of proportionality, is described by the following differential equation.

For this particular function it's known that the population is in the form to represent the dogs. Since the virus spreads based on the interactions between the dogs who have the virus and those who have not yet contracted it, the population will be the product of the number of dogs with the virus and those without the virus.

Therefore the differential equation becomes,

By the cauchy-peano theorem, for , as long as is continuous on a closed rectangle around our starting point, we have local existence. All three functions are continuous everywhere, so they enjoy local existence at every starting point.

We can show that the solutions to differential equations are unique by showing that is Lipschitz continuous in y. If is continuous, then this will suffice to show the Lipschitz continuity.

Note that the first and third equations are continuous for all y and t, but that the second is not continuous when . More concretely, when , both the equation and the equation would satisfy the differential equation.

By the cauchy-peano theorem, for , as long as is continuous on a closed rectangle around our starting point, we have local existence. All three functions are continuous everywhere, so they enjoy local existence at every starting point.

We can show that the solutions to differential equations are unique by showing that is Lipschitz continuous in y. If is continuous, then this will suffice to show the Lipschitz continuity.

Note that the first and third equations are continuous for all y and t, but that the second is not continuous when . More concretely, when , both the equation and the equation would satisfy the differential equation.

By definition, an autonomous differential equation does not depend explicitly on the independent variable. An autonomous differential equation will take the form

By definition, an autonomous differential equation does not depend explicitly on the independent variable. An autonomous differential equation will take the form