Exact Equation - Arbitrary Constant

Category: Differential Equations, Integral Calculus, Differential Calculus

"Published in Newark, California, USA"

Solve for the particular solution for

when x = 0, y = 2.

Solution:

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

Let

then

Let

then

Since

then the given equation is Exact Equation. The solution for the above solution 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

To solve for T(y), integrate on both sides of the equation with respect to y, as follows

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,

The general solution of the equation is

To solve for the value of C, substitute x = 0 and y = 2 to the above equation, we have

Therefore, the particular solution of the equation is

Indeterminate Form - One Raised Infinity

Category: Differential Calculus

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation

Substitute the value of x to the above equation

One raised to infinity is also another type of Indeterminate Form and it is not accepted as a final answer in Mathematics. We know that one raised to any number is always equal to one but in this case, one raised to infinity is not always equal to one, that's why 1^∞ is also an Indeterminate Form. In this type of Indeterminate Form, we cannot use the L'Hopital's Rule because the L'Hopital's Rule is applicable for the Indeterminate Forms like 0/0 and ∞/∞. Since the above equation involves exponential function, we can rewrite the equation as follows

let

Take natural logarithm on both sides of the equation

Substitute the value of x to the above equation

Since the Indeterminate Form is ∞∙0, we have to rewrite the equation again as follows

Substitute the value of x to the above equation

Since the Indeterminate Form is 0/0, then we can now use the L'Hopital's Rule as follows

Substitute the value of x to the above equation

Take inverse natural logarithm on both sides of the equation

Therefore,

Finding Equation - Parabola

Category: Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a parabola with horizontal axis, vertex on y axis, and passing through the points (2, 4) and (8, -2).

Solution:

To illustrate the problem, let's plot all the given items and sketch the parabola in the rectangular coordinate system as follows

Photo by Math Principles in Everyday Life

As you can see in the figure, we can have two equations of parabola based on the given items in the problem. Since the axis of the parabola is horizontal and it opens to the right, the equation of a parabola in standard form is

The vertex of a parabola is located in y-axis, the coordinates of the vertex is (0, k). The above equation becomes

If (2, 4) is one of the points of a parabola, substitute the values of x and y to the above equation

If (8, -2) is one of the points of a parabola, substitute the values of x and y to the above equation

Equate

Multiply both sides of the equation by 32 and solve for the value of k

Divide both sides of the equation by 3

Equate each factor to zero and solve for the value of k

If 

then

Solve for the value of a, we have

Therefore, the equation of a parabola in standard form is

If

then

Solve for the value of a, we have

Therefore, the equation of a parabola in standard form is