Using integrating factors:

Once we use this integrating factor, we can reduce the equation to:

Integrating both sides: 

Since you may or may not know how to solve  order Differential equations, you can always plug in the answers and see if it is correct. 

Let's start with : 

If you try all the other function, they will solve the differential equation. 

This involves solving a differential equation for . 

Firstly, we determine the value of :

At 

This means our equation is now:

To determine the difference of  from , we take the definite integral:

The derivative of the trigonometric function  is .

Therefore, the correct answer is 

.

To find the derivative of the function, you must use the power rule 

.

Using this rule, the correct answer 

 is obtained.

To take the derivative of the first two terms, you use the power rule 

.

To take the derivative of sin(x), there is the identity that,

 .

Using all of the rules and applying, 

.

Solving a differential equation tends to be quite simple if it is separable. A separable equation is one that can be rewritten in the following form 

.  

From here we can manipulate the equation 

and then you can integrate both sides.  

We can rewrite the equation as 

and we therefore see that it is separable.  

Next we manipulate it to get the the y terms on the left hand side and the x terms on the right hand side like the following 

.  

The next step is to integrate both sides 

.  

This gives us 

,

since the c's are just arbitrary constants we can combine them to get 

.  

Since we want to write this as an equation in terms of y we will apply e to both sides.  

Therefore we get 

.  

Since the problem gives us an initial condition we solve for c.  

Therefore 

so c=4.  

Thus the solution is 

.

When given a first order differential equation to solve, one of the first things to check is that it is homogenous.Homogenous equations are ones in which, given

, 

 given any t.

If an equation is homogenous you can do a substitution of  and this new equation will be separable.  

The first step is to check if the following equation is homogenous, 

.  

Next we show that the equation is homogenous

.

 So we now  replace y with xz. This gives us the following new equation:

.  

This is a separable equation which we manipulate and integrate both sides 

which leads to 

.  

Thus we get the equation 

,

then plugging y back in we get 

.  

So our family of equations is 

yet plugging in the given condition gives us that c=0.  

Thus the final solution 

.

Recall that a function has critcal points where its first derivative is equal to zero or undefined.

So, given , we need to find v'(t)

Where is this function equal to zero? t=0 for one. We can find the others, but we really just need one. Plug in t=0 into our original equation to find the point (0,0) Which is in this case a saddle point.

A local max will occur when the function changes from increasing to decreasing. This means that the derivative of the function will change from positive to negative.

First step is to find the derivative.

Find the critical points (when  is  or undefined).

Next, find at which of these two values  changes from positive to negative. Plug in a value in each of the regions into .

The regions to be tested are ,, and .

A value in the first region, such as , gives a positive number, and a value in the second range gives a negative number, meaning that  must be the point where the max occurs.

To find what the  coordinate of this point, plug in  in to , not , to get .