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

 

Solving Equations - Homogeneous Functions, 8

Category: Differential Equations

"Published in Vacaville, 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? An exponential function is a combination of x and y in the function 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