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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 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
Next, we need to solve for the value of A, B, and C by partial fractions as follows
Equate the coefficients for u², we have
Equate the coefficients for u, we have
Equate the coefficients for u0, we have
Then it follows that
Substitute the value of A, B, and C to the above 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
