Question

In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:
$straight y space equals space straight x space sinx$ : $xy apostrophe space equals space straight y plus straight x square root of straight x squared minus straight y squared end root$ $left parenthesis straight x not equal to 0 space and space straight x greater than straight y space or space straight x less than negative straight y right parenthesis$

Solution

$straight y space equals space straight x space sinx$ ...(1)
$therefore space space space space space straight y apostrophe space equals space straight x space cosx space plus space sinx rightwards double arrow space space space xy apostrophe space equals space straight x squared cosx plus straight x space sinx rightwards double arrow space space space xy apostrophe space equals straight x squared square root of 1 minus sin squared straight x end root space plus space straight x space sinx rightwards double arrow space space xy apostrophe space equals space straight x squared square root of 1 minus straight y squared over straight x squared end root plus straight y space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets rightwards double arrow space space xy apostrophe space equals space straight x square root of straight x squared minus straight y squared end root plus straight y rightwards double arrow space space straight x space straight y apostrophe space equals space straight y plus straight x square root of straight x squared minus straight y squared end root$
$therefore space space space straight y space equals space straight x space sinx$ is a solution of $xy apostrophe space equals straight y plus straight x square root of straight x squared minus straight y squared end root.$

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Vector Algebra

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0

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Vector Algebra

In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation: :