Differential Equations

Solve the Initial Value Problems (IVP) below:


Solution

This is an homogeneous differential equation and it is separable. First, we will divide everything by sin x

Now, we will make the following change:

in the equation [1]  above. Then,

Working with this equation a little bit we get:

Integrating both sides we get:

where

Using the IVP value given in the problem we will have:

Making the necessary calculations, we will get:

Using this result in [3] above, we finally arrive at:

Which is the equation we’ve been looking for.


Solution

This is also an Homogeneous Differential Equation and it is also separable. Repeating the same steps taken above to separate the variables, we will get the following equation:

where

then,

Integrating, we get:

Which implies that:

Using the IVP value in [5] above, we get:

substituting [6] in [5] we finally have:

Which is a particular solution for the given Differential Equation.