Not all differential equations take a standard form which can be solved used a standard result. Most differential equations can only be solved numerically on a computer, to a certain precision, which is always a compromise with the time and computing power available and the required precision. Finding the transformation to use is a matter of experience and intuition and recognition. If you are given the transformation then it is much simpler.
The equation(1) may be transformed into a simpler form using the transformation y=vx
On substituting these expressions into (1) the equation becomes
The final step is to divide bybecause it is a common factor, but we can only do this if we include the condition thatelse we would be dividing by zero. We obtain
This is a nonhomogenous linear equation which can easily be solved using standard techniques. The solution, once found is transformed back into the original variables x and y.
The equation(2) may be transformed using the transformation
The equation (2) becomes
This can be directly integrated and transformed back into the original variables x and y to solve the original equation.