Solving Equations - Homogeneous Functions, 3

Category: Differential Equations

"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, 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.