Solving Equations - Homogeneous Functions, 9

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 and second term are the 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 

  

Take the inverse natural logarithm on both sides of the equation, we have

But

Hence, the above equation becomes  

where D = C².

Therefore, the general solution is

 

Separation of Variables, 16

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 - x)y² 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, 15

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 x³y³ as follows

Integrate both sides of the equation, we have 

where D = 2C.

Therefore, the general solution is