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.

这个probl的第二部分em 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 variableand 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 variableand all its derivatives have a power involving one and all the coefficients depend ontherefore, this differential equation is linear.

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

By the cauchy-peano theorem, for, as long asis 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 thatis Lipschitz continuous in y. Ifis 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 equationand the equationwould satisfy the differential equation.

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.

这个probl的第二部分em 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 variableand 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 variableand all its derivatives have a power involving one and all the coefficients depend ontherefore, this differential equation is linear.

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

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

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.

这个probl的第二部分em 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 variableand 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 variableand all its derivatives have a power involving one and all the coefficients depend ontherefore, this differential equation is linear.

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

This equation is third order since that is the highest order derivative present in the equation.

This is equation in linear becauseand derivatives appear to the first power only.anddo not affect the linearity.

First identify what is known.

The general function is,

The initial value is six in mathematical terms is,

From here, substitute in the initial values into the function and solve for.

Finally, substitute the value found forinto the original equation.

So this is a separable differential equation, but it is also subject to an initial condition. This means that you have enough information so that there should not be a constant in the final answer.

You start off by getting all of the like terms on their respective sides, and then taking the anti-derivative. Your pre anti-derivative equation will look like:

Then taking the anti-derivative, you include a C value:

Then, using the initial condition given, we can solve for the value of C:

Solving for C, we get

which gives us a final answer of:

We haveso thatand. Solving for y,

and

which we can write becauseis just another arbitrary constant.

Plugging in our initial value, we haveleaving us with a final answer of.

Note, this type of equation pops up frequently in the course and is potentially good to just memorize. For, we have

So this is a separable differential equation with a given initial value.

To start off, gather all of the like variables on separate sides.

Then integrate, and make sure to add a constant at the end

To solve for y, take the natural log, ln, of both sides

Be careful not to separate this, a log(a+b) can't be separated.

Plug in the initial condition to get:

So raising e to the power of both sides:

Solving for C:

giving us a final answer of: