Solving Equations - Homogeneous Functions, 5

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

where D = 1/C.

Therefore, the general solution is

 

Variable Separation, 13

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 x²y 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, 12

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 (1 + y²)(1 + x²) as follows

Integrate both sides of the equation, we have

Apply the inverse trigonometric identities formula at the left side of the equation, we have

Take tangent on both sides of the equation, we have

where D = tan C.

Therefore, the general solution is