Form the differential equation of the family | vedclass

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Question

(B) Given the family of lines: $y = mx + \frac{4}{m}$  $(i)$Differentiating both sides with respect to $x$:$\frac{dy}{dx} = m$Substitute $m = \frac{dy

Vedclass · Accepted Answer

$x\left(\frac{dy}{dx}\right)^{2} - y\frac{dy}{dx} + 4 = 0$

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(B) Given the family of lines: $y = mx + \frac{4}{m}$  $(i)$Differentiating both sides with respect to $x$:$\frac{dy}{dx} = m$Substitute $m = \frac{dy}{dx}$ into equation $(i)$:$y = x\left(\frac{dy}{dx}\right) + \frac{4}{\frac{dy}{dx}}$Multiply the entire equation by $\frac{dy}{dx}$ to clear the denominator:$y\left(\frac{dy}{dx}\right) = x\left(\frac{dy}{dx}\right)^{2} + 4$Rearranging the terms to form the differential equation:$x\left(\frac{dy}{dx}\right)^{2} - y\frac{dy}{dx} + 4 = 0$Thus,the required differential equation is $x\left(\frac{dy}{dx}\right)^{2} - y\frac{dy}{dx} + 4 = 0$.

Form the differential equation of the family of lines $y = mx + \frac{4}{m}$ by eliminating the arbitrary constant $m$.

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