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