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Question
Form the differential equation from the following primitive where constants are arbitrary:
y2 = 4ax
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Solution
The equation of family of curves is \[y^2 = 4ax.................(1)\]
where a is an arbitrary constant.
This equation contains only one arbitrary constant, so we shall get a differential equation of first order.
Differentiating equation (1) with respect to x, we get
\[2y\frac{dy}{dx} = 4a\]
\[ \Rightarrow \frac{y}{2}\frac{dy}{dx} = a .................\left( 2 \right)\]
Putting the value of a in equation (1), we get
\[y^2 = 4\frac{y}{2}\frac{dy}{dx}x\]
\[ \Rightarrow y = 2x\frac{dy}{dx}, \]
It is the required differential equation.
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