Question
Form the differential equation corresponding to $\text{y}=\text{e}^{\text{mx}}$ by eliminating m.
$\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 $\frac{1}{\text{x}}\int\text{y}=\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}$ $\Rightarrow\text{x}\frac{\text{dy}}{\text{dx}}=\text{y}\int\text{y}$Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
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$\int\sin^{-1}\Big(\frac{2\text{x}}{1+\text{x}^2}\Big)\text{dx}$
f(x) = (x - 1)2 + 2 on R.