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.

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.

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.

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

This is equation in linear because  and derivatives appear to the first power only.  and  do 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 for  into 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 have  so that  and . Solving for y,

 and 

which we can write because  is just another arbitrary constant.

Plugging in our initial value, we have  leaving 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:

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

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

Notice that when you divide sec(y) to the other side, it will just be cos(y),

and the csc(x) on the bottom is equal to sin(x) on the top. 

  Integrating, we get:

 so we can plug pi/4 into both x and y:

  this gives us a C value of 

In order to solve for y, we just need to take the arcsin of both sides:

So if you rearrange this equation, you will arrive at a separable differential equation by adding the  to the other side:

   Now, to solve this, multiply the dx to the other side and take the anti-derivative:

Then, after the anti-derivative, make sure to add the constant C:

  Now, plug in the initial condition that y(0)=0, which will give you a C=0 as well. Then just take the square root, and you arrive at:

 Then, to get the correct answer, simplify by factoring out an  and pulling it outside of the square root to get: