Question

In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:
$straight y space equals space square root of straight a squared minus straight x squared end root space comma space straight x space element of space left parenthesis negative straight a comma space straight a right parenthesis space colon space space space space space space straight x plus straight y space dy over dx space equals space 0 space space space left parenthesis straight y not equal to 0 right parenthesis$

Solution

$straight y space equals space square root of straight a squared minus straight x squared end root$ ...(1)
$therefore space$ $straight y apostrophe space equals space fraction numerator negative 2 straight x over denominator 2 square root of straight a squared minus straight x squared end root end fraction$
$rightwards double arrow space space space space straight y apostrophe equals negative fraction numerator straight x over denominator square root of straight a squared minus straight x squared end root end fraction$
$rightwards double arrow space space space space space space space space space dy over dx equals negative straight x over straight y space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets rightwards double arrow space space space space space space straight y dy over dx space equals space minus straight x rightwards double arrow space space space space straight x plus straight y dy over dx space equals space 0 space therefore space space space straight y space equals space square root of straight a squared minus straight x squared end root space is space straight a space solution space of space straight x plus straight y dy over dx space equals space 0.$

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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: