# Calculus 1 : How to find solutions to differential equations

## Example Questions

### Example Question #341 : Equations

What is the derivative of   at ?

Explanation:

Here you must use the chain rule,

and the rule for natural log,

.

First, you get

,

then using the chain on , you get

You must then remember the , which gives you

as the final derivative.

Plugging in  gives .

### Example Question #342 : Equations

Find  for the equation:

Explanation:

Take the derivative of each term in the equation twice: with respect to  and then with respect to . When taking the derivative with respect to one variable, treat the other variable as a constant.

For the function

The derivative is then

Now bring  and  terms to opposite sides of the equation:

Now rearraging variables gives :

### Example Question #343 : Equations

Find  for the following equation:

Explanation:

Given the function:

We'll be taking the derivative with respect to the variables  and .

Beginning with the left side of the equation, only the  variable appears and the derivative is:

Note that

Now for the right side of the equation, both  and  appear, so we'll utilize the chain rule giving the derivative:

Combining these gives the derivative of the original equation:

Since we're looking for , gather  terms on one side of the equation and  terms on the other side:

From there, separate terms once more to find :

### Example Question #104 : Differential Equations

Find  for the equation:

Explanation:

For this problem, knowledge of the following derivatives is necessary:

To take the derivative of the equation

Let's begin with the left side. For each term, we'll take the derivative with respect to both variables  and , treating the other variable as just a constant when we do so. The derivative of the left side is thus

Now moving to the right side, the derivative is:

Notice how since the  term has no  term, when we take the derivative with respect to  we just get zero, since we're treating the  as a constant.

Now we have the derived equation:

Bring  and  terms to opposite sides of the equation:

Now we can once more rearrange variables to find :

### Example Question #341 : Equations

Find  for the equation:

Explanation:

We'll be taking the derivative of each term in the equation

with respect to both  and . When taking the derivative for one variable, treat the other variable as a constant:

Note that

Notice how  is treated as a constant when taking the derivative with respect to  and so goes to zero.  The same happens when taking the derivative of  with respect to

Now bring  and  terms to opposite sides of the equation:

Rearrange terms once more to find :

### Example Question #106 : Differential Equations

Find  for the equation:

Explanation:

For this problem, note that:

To approach this problem, we'll take the derivative of each term twice: with respect to  and then with respect to . When taking the derivative with respect to one variable, treat the other variable as a constant.

For the equation

The derivative is then

Now, bring  and  terms to opposite sides of the equation:

Finally, rearrange terms once more to get :

### Example Question #107 : Differential Equations

Find  for the equation:

Explanation:

For this problem, note that:

Take the derivative of each term in the equation twice: with respect to  and then with respect to . When taking the derivative with respect to one variable, treat the other variable as a constant.

For the function

The derivative of each side is

Now move  and  terms to opposite sides of the equation:

Finally rearrange variables to get :

### Example Question #108 : Differential Equations

Find  for the equation:

Explanation:

For this problem, note that:

Take the derivative of each term in the equation twice: with respect to  and then with respect to . When taking the derivative with respect to one variable, treat the other variable as a constant.

For the function

The derivative is then

Now bring  and  terms to opposite sides of the equation:

Now rearraging variables gives :

### Example Question #109 : Differential Equations

Find  for the equation:

Explanation:

For this problem, note that:

Product rule:

Take the derivative of each term in the equation twice: with respect to  and then with respect to . When taking the derivative with respect to one variable, treat the other variable as a constant.

For the function

The derivative is then

Remember to utilize the chain rule!

Now bring  and  terms to opposite sides of the equation:

Now rearraging variables gives :

### Example Question #110 : Differential Equations

Find  for the equation:

Explanation:

Note that:

Product Rule:

Take the derivative of each term in the equation twice: with respect to  and then with respect to . When taking the derivative with respect to one variable, treat the other variable as a constant.

For the function

The derivative is then found using the product rule to be:

Notice how the chain rule needs to be utilized an additional time when taking the derivative of the  term with respect to .

Now bring  and  terms to opposite sides of the equation:

Now rearraging variables gives :