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Thursday, July 17, 2014

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