Form the differential equation corresponding to y=e^ mx by elimin… - Vidyadip

Question

Form the differential equation corresponding to $\text{y}=\text{e}^{\text{mx}}$ by eliminating m.

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Answer

The equation of the family of curves is$\text{y}=\text{e}^{\text{mx}}...(1) $
where m is a parameter.
This equation contains only one parameter, so we shall get a differential equation of first order. Differentiating equation (1) with respect to x, we get
$\frac{\text{dy}}{\text{dx}}=\text{me}^\text{mx}$ $\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{my}$ $\Rightarrow\text{m}=\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}\ ...(2)$ Now, from equation (1), we get $\int\text{y}=\text{Ine}^{\text{mx}}$ $\Rightarrow\int\text{y}=\text{mx Ine}$ $\Rightarrow\int\text{y}=\text{mx}$ $\Rightarrow\text{m}=\frac{1}{\text{x}}\int\text{y}$ Compairing equation (2) and (3), we get $\frac{1}{\text{x}}\int\text{y}=\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}$ $\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}\int\text{y}$

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