Orthogonal And Oblique Trajectories

Question 1

How do you find the orthogonal trajectory of a given family?

Answer 1

1. Differentiate the function implicitly (not partially!) with respect to x.
2. Eliminate ‘c’ by equating it to the variables in the original function and substituting the result in the differential equation.
3. Re-write the differential equation so it equates to the negative inverse of the function (Don’t flip the dy/dx!)
4. Variable separable and integrate.
5. Write the equation as a function of x and y equalling c.

Question 2

What does it mean for two families to be orthogonal to each other?

Answer 2

This implies that m1 • m2 = -1
That is, the product of the derivatives of both families gives -1.

Question 3

How do you find the oblique trajectories?

Answer 3

1. Differentiate the equation.
2. Eliminate ‘c’ by rearranging the given equation, equating ‘c’ to the other variables x and y.
3. Slot in the function of x and y into: [f(x,y) + tanα] ÷ [1–f(x,y)tanα]
4. Solve the differential equation: it may be easily done by variable separable, or in need of solving homogenously (y =vx)
5. Equate to zero.