Solving Logarithmic Equation, 3

Category: Algebra, Trigonometry

"Published in Newark, California, USA"

Solve for x:

Solution:

Consider the given equation above

Did you notice that the bases of the logarithmic functions are different? Well, we have to convert all the logarithmic functions into the same base first. Let's convert all the logarithmic functions into base 2 as follows

for

for

for

Hence, the given equation becomes

Take out their common factor, we have

Since the terms inside the bracket are all coefficients, then we can eliminate the coefficient since the right side of the equation is zero.

Take inverse logarithm on both sides of the equation

Take inverse tangent on both sides of the equation

Therefore, the answer is

I would like to thank Mr. Bilomba Nkita Leonard, Educator of Mathematics and Physical Sciences at Northwest Department of Education in South Africa who posted this problem at LinkedIn. 

Solving Equations - Homogeneous Functions, 2

Category: Differential Equations, Integral Calculus

"Published in Newark, California, USA"

Find the general solution for

Solution: 

Consider the given equation above

Did you notice that the given equation cannot be solved by separation of variables? The first term is a combination of x and y in the group and there's no way that we can separate x and y. 

This type of differential equation is a homogeneous function. Let's consider this procedure in solving the given equation as follows

Let

so that

Substitute the values of y and dy to the given equation, we have

The resulting equation can now be separated by separation of variables as follows

Integrate on both sides of the equation, we have

But

Hence, the above equation becomes

Therefore, the general solution is

 

Variable Separation, 6

Category: Differential Equations, Integral Calculus

"Published in Newark, California, USA"

Find the general solution for 

Solution:

Consider the given equation above

Transpose xy to the right side of the equation, we have

Arrange the above equation by separation of variables, we have

Integrate on both sides of the equation, we have

Take the inverse natural logarithm on both sides of the equation

where K = e^C. Therefore, the general solution is