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Question
Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
y = ax3
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Solution
The equation of family of curves is \[y = a x^3.........(1)\]
where `a` is a parameter.
As this equation has only one arbitrary constant, so we shall get a differential equation of first order.
Differentiating (1) with respect to x, we get
\[\frac{dy}{dx} = 3a x^2 \]
\[ \Rightarrow \frac{dy}{dx} = 3 \times \frac{y}{x^3} \times x^2 .........\left[\text{Using}\left( 1 \right) \right]\]
\[ \Rightarrow x\frac{dy}{dx} = 3y\]
It is the required differential equation.
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