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Friday, July 11, 2014

Solving Equations - Homogeneous Functions, 6

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  





But



Hence, the above equation becomes 







where D = 2C.

Therefore, the general solution is

 

Thursday, July 10, 2014

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

 

Wednesday, July 9, 2014

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