Solving Exact Equations

Category: Differential Equations

"Published in Newark, California, USA"

Solve for the solution for

Solution:

The first thing that we have to do is to check the above equation if it is exact equation or not as follows

Let 

then 

Let 

then

Since

then the given equation is Exact Equation. The solution for the above equation is F = C. Consider the given equation

Let 

and

Integrate the partial derivative of the first equation above with respect to x, we have

Since we are integrating the partial derivatives, then another unknown function, T(y) must be added.  If 

then

To solve for T'(y), equate

Since the arbitrary constant is already included in F = C, then we don't have to add the arbitrary constant in the above equation. Therefore,

Therefore, the general solution is

Indeterminate Form - Infinity Minus Infinity

Category: Differential Calculus, Algebra

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Evaluate the limit for

Solution:

Consider the given equation, substitute the value of x, we have

Since the answer is ∞ - ∞ which is also another type of Indeterminate Form, it is not accepted in Mathematics as a final answer. In this type of Indeterminate Form, you cannot use the L'Hopital's Rule because the L'Hopital's Rule is applicable for the Indeterminate Forms like 0/0 and ∞/∞. In this case, let's rewrite the given equation as follows

Substitute the value of x to the above equation

Since the Indeterminate Form at this time is 0/0, we can now apply the L'Hopital's Rule

As you notice that when you substitute the value of x to the above equation, the answer will be 0/0 again. If you will apply the L'Hopital's Rule to the above equation, the equation will be more complicated and the answer will be 0/0 again. In this case, we have to rewrite the above equation as follows

We cannot apply the L'Hopital's Rule for the above equation since the numerator will be ∞ - ∞ when x equals ∞. If you will rewrite the above equation, the trend of rewriting the equation will be the same over and over and it will be more complicated as well.

But don't worry, we have a special solution for the exponential and logarithmic functions whose Indeterminate Forms are ∞ - ∞. Let's consider this one

Let 

so that

If x equals ∞, the above equation will be

Since the Indeterminate Form is ∞/∞, then we can apply the L'Hopital's Rule for the above equation as follows

Since

then it follows that

Therefore,

Integration Procedure - Parts

Category: Integral Calculus

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Find the integral for

Solution:

As you notice that there are two functions in the given equation but there's no differentials for the two functions. In this case, we have to use the integration procedure which is Integration by Parts. 

Let

By using Integration by Parts,

Substitute the values of u, v, du, and dv to the above equation

Therefore,