Question

Will someone give me explanation of this question below??

$show \: that \: function \: y = ax+ {2a}^{2} \: is \: a \: solution \: of$
$the \: differential \: equation$
$2( \frac{dy}{dx})^{2} + x ( \frac{dy}{dx} ) - y = 0$
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Answer 1

If y = ax + 2a², then its derivative is dy/dx = a. Substitute y and dy/dx into the given equation and check if it results in an identity:

2 (dy/dx)² + x dy/dx - y = 0

2a² + ax - (ax + 2a²) = 0

0 = 0

This is of course true, so y = ax + 2a² is indeed a solution to the given equation.