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Find the equation of a curve whose slope at any point is equal to y/(y - x) and which passes through the point (-1, 2).
Solution:
If you want get the slope of a curve at any point whose variables are second degree, third degree, fourth degree, and so on, then you need to take the first derivative of a curve with respect to an independent variable which is x in common. In this case, the given slope of a curve can be written as
The above equation cannot be solved by Separation of Variables because dy is multiplied by (y - x) and there's no way to separate the variables. This type of equation is Homogeneous Function. We have to substitute a variable with another variable first in order to solve the given equation by Separation of Variables as follows:
Let
so that
Substitute the value of y and dy to the above equation, we have
The above equation can now be solved by Separation of Variables as follows
Since the right side of the equation is a rational function in which the denominator has two factors, then it must be expressed into partial fractions:
Multiply both sides of the equation by their Least Common Denominator (LCD) which is u(2 - u), we have
Equate their u:
Equate their constants:
Substitute the value of A to other equation in order to solve for the value of B
Substitute the values of A and B to the original equation, we have
Integrate on both sides of the equation
Apply the laws of logarithm to the above equation
Take the inverse natural logarithm on both sides of the equation
Square on both sides of the equation
where C2 = D
We know that
Substitute the value of u to the above equation, we have
But x = -1 and y = 2 from the given point of a curve which is (-1, 2), then the value of D will be equal to
Therefore, the equation of a curve that passes thru the point (-1, 2) is
