Question

The differential equation representing the family of curves $straight y squared space equals space left parenthesis straight x plus square root of straight c right parenthesis$ , where c > 0, is a parameter, is of order and degree as follows:

order 1, degree 2

order 1, degree 1

order 1, degree 3

order 2, degree 2

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Solution

order 1, degree 3

straight y squared space equals 2 straight c space left parenthesis straight x space plus square root of straight c right parenthesis space..... space left parenthesis straight i right parenthesis 2 yy apostrophe space equals space 2 straight c.1 space or space yy apostrophe space equals space straight c space.... space left parenthesis ii right parenthesis rightwards double arrow space straight y squared space equals space 2 yy apostrophe space left parenthesis straight x plus square root of yy apostrophe end root right parenthesis space left square bracket on space putting space value space of space cfrom space left parenthesis ii right parenthesis space and space left parenthesis straight i right parenthesis right square bracket On space simplifying comma space we space get space left parenthesis straight y minus 2 xy apostrophe right parenthesis squared space equals space 4 yy apostrophe cubed space left parenthesis iii right parenthesis Hence space equation space left parenthesis iii right parenthesis space is space of space order space 1 space adn space degree space 3.

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