Question
Form the differential equation corresponding to $\text{y}=\text{e}^{\text{mx}}$ by eliminating m.
✓
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
$\frac{1}{\text{x}}\int\text{y}=\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}$
$\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}\int\text{y}$
$\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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