Homogeneous Functions - Arbitrary Constant

Category: Differential Equations, Integral Calculus

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Find the particular solution for 

when 

               

Solution:

If you examine the given equation, it is a differential equation because it has dy and dx in the equation. The type of equation is Homogeneous because the functions and variables cannot be separated by Separation of Variables. There's a method to solve the Homogeneous Functions. Consider the given equation

Let y = vx

    dy = vdx + xdv

Substitute y and dy to the above equation, we have

The above equation can now be separated by Separation of Variables. Arrange the equation according to their variables

Integrate both sides of the equation

Take the inverse natural logarithm on both sides of the equation

but y = vx and v = ^y/_x, 

To solve for C, substitute the following:

to the above equation, we have

Therefore,

  

Dividing Rational Fractions

Category: Algebra

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Perform the indicated operations and simplify

Solution:

This is a division of a rational fraction with another rational fraction. As a rule in Mathematics that we need to get the reciprocal of the divisor first and then perform the multiplication as follows

Factor all the polynomials in the numerator and denominator. Do this by trial and error so that the middle term of a polynomial is matched.

Simplify the above equation

Therefore, the answer is 

Solving Parallelogram Equation

Category: Analytic Geometry

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Given a parallelogram with vertices A(- 2, - 1), B(4, 2), C(7, 7) and D(1, 4).

a. Find the equation of a parallelogram as a function of x.

b. Prove that the given parallelogram is real a parallelogram.

Solution:

The first that we have to do is to plot the vertices of a parallelogram and draw the figure as well.

Photo by Math Principles in Everyday Life

a. To get the equation of a parallelogram as a function of x, we need to get the equations of the sides of the parallelogram using the two point form. 

For AB, substitute the values of points A and B

For BC, substitute the values of points B and C

For CD, substitute the values of points C and D

For DA, substitute the values of points D and A

Therefore, the equation of a parallelogram as a function of x is 

b. Since the slopes of the opposite sides of a parallelogram are equal, then the given parallelogram is real a parallelogram.

                        m_AB = m_CD = ½

                        m_BC = m_DA = ⁵/₃