Solving Equations - Homogeneous Functions, 13

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 algebraic functions 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 

But

Hence, the above equation becomes 

 
Therefore, the general solution is

 

Separation of Variables, 20

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 y(1 + e^2x) 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

 

Separation of Variables, 19

Category: Differential Equations

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above 

Transfer all the terms from the right side of the equation to the left side of the equation and group as follows

In order to separate dx and dy from other variables, divide both sides of the equation by x²y² as follows
 

 

 

   
Integrate both sides of the equation, we have

Therefore, the general solution is