Variable Separation, 8

Category: Differential Equations

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above 

In order to separate dx and dy from other variables, divide both sides of the equation by y²(x² + 1) as follows 

Integrate both sides of the equation, we have

Take the inverse natural logarithm on both sides of the equation, we have

Therefore, the general solution is

 

Variable Separation, 7

Category: Differential Equations

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above

Since the grouped terms consist of only one variable, then we can divide both sides of the equation by (y + 2)(x - 2) so that we can separate dx and dy from other variables as follows

Integrate both sides of the equation, we have

Take the inverse natural logarithm on both sides of the equation, we have

Therefore, the general solution is

where K = C + 4.

Indeterminate Form - Infinity Over Infinity, 4

Category: Differential Calculus, Algebra

"Published in Vacaville, California, USA"

Evaluate

Solution:

To get the value of a given function, let's substitute the value of x to the above equation, we have 

Since the answer is ∞/∞, then it is an Indeterminate Form which is not accepted as a final answer in Mathematics. We have to do something first in the given equation so that the final answer will be a real number, rational, or irrational number.

Method 1:

Since the answer is Indeterminate Form, then we have to divide both sides of the fraction by a term with the highest degree which is x² and simplify the given equation as follows

Substitute the value of x to the above equation, we have

Therefore,

Method 2:

Another method of solving Indeterminate Form is by using L'Hopital's Rule. This is the better method especially if the rational functions cannot be factored. L'Hopitals Rule is applicable if the Indeterminate Form is either 0/0 or ∞/∞. Let's apply the L'Hopital's Rule to the given function by taking the derivative of numerator and denominator with respect to x as follows

Substitute the value of x to the above equation, we have

  
Again, apply the L'Hopital's Rule to the above equation, we have

Did you notice that the final equation is similar to the original equation? If you will continue this process, there will be an endless repetition of the process. Instead, let's consider the equation after the first application of L'Hopital's Rule as follows

   
Let's rewrite the right side of the equation by including the numerator into the radical at the denominator as follows

Perform the division of polynomials inside the radical, we have

Substitute the value of x to the above equation, we have

Therefore,