Find the differential equation of the family of lines passing through the origin.

Solution

Consider the equation, y = mx, where m is the parameter.
Thus, the above equation represents the family of lines which pass through the origin.
y = mx ....(1)
$rightwards double arrow space space straight y over straight x space equals space straight m space... left parenthesis 2 right parenthesis$
Differentiating the above equation (1) which respect to x,
$straight y space equals space mx dy over dx space equals space straight m space cross times space 1 space space rightwards double arrow space space dy over dx equals straight m space space space rightwards double arrow space dy over dx space equals space straight y over straight x space left square bracket because space from space equation space left parenthesis 2 right parenthesis right square bracket space space space space rightwards double arrow dy over dx minus space straight y over straight x space equals space 0 space$
Thus we have eliminated the constant, m.
The required differential equation is
$dy over dx minus straight y over straight x space equals 0$

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Vector Algebra

Find the differential equation of the family of lines passing through the origin.

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