Solving Equations - Homogeneous Functions, 14

Category: Differential Equations

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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  

 

 

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

But

Hence, the above equation becomes

where D = C³.

Therefore, the general solution is