Question

In the following, verify that the given functions, (explicit or implicit) is a solution of the corresponding differential equation:
$xy space equals space log space straight y space plus space straight C$ $colon space space straight y apostrophe space equals space fraction numerator straight y squared over denominator 1 minus straight x space straight y end fraction left parenthesis space xy not equal to 1 right parenthesis$

Solution

$xy space equals space log space straight y space plus space straight C$
Differentiate w.r.t.x, we get
$xy apostrophe plus straight y.1 space equals space space 1 over straight y. straight y apostrophe space plus space 0$
or $xy apostrophe plus straight y space equals space fraction numerator straight y apostrophe over denominator straight y end fraction$
$rightwards double arrow space space space xyy apostrophe plus straight y squared space equals space straight y apostrophe rightwards double arrow space space space space space space space space space space space straight y squared space equals space straight y apostrophe minus xyy apostrophe rightwards double arrow space space space space space space space space space straight y squared space equals space straight y apostrophe left parenthesis 1 minus xy right parenthesis rightwards double arrow space space space space space space space space straight y apostrophe space equals space fraction numerator straight y squared over denominator 1 minus xy end fraction$
$therefore space space xy space equals space logy plus straight C space is space straight a space solution space of space straight y apostrophe space equals space fraction numerator straight y squared over denominator 1 minus xy end fraction.$

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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

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