collection of problems in integration, differentiation, areas, volumes, trigonometry, matrices, limits

Calculus
problems on integration ------------------------ integration

problems on application of integration like
area, volume, arc length etc ------------------------integration application

problems on differential equations ------differential equation

problems on differentiation------------------------ differentiation

problems on limits ------------------limits

problems on matrices --------------matrices

problems on trigonometry ------trigonometry


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orthogonal trajectory of y = (k/x)

find the orthogonal trajectory of y = (k/x)

first we find the diff. equation of the given family
y=(k/x) -----------(1)
diff. w.r.t. x
y' = -k / (x²)---------(2)

eliminating k , divide (2) by (1)

y' / y = -1/x ---------(3)

(3) is the diff. equation of the given family.
replace y with -1/y' in (3) to get the diff. equation of the orthogonal trajectory

diff. equation of the orthogonal trajectory is

-yy' = -1/x
y(dy/dx) = 1/x

ydy =xdx
integrating and simplifying, we see that the orthogonal trajectory is

x² - y² = C


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