Question

if xy=c^2 proove that x^2 dy/dx+c^2=0

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{xy=c^2}$

$\underline{\textbf{To prove:}}$

$\mathsf{x^2\;\dfrac{dy}{dx}+c^2=0}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{xy=c^2}$

$\textsf{Using product rule, differentiate with respect to 'x'}$

$\mathsf{x\;\dfrac{dy}{dx}+y.1=0}$

$\mathsf{x\;\dfrac{dy}{dx}+y=0}$

$\mathsf{x\;\dfrac{dy}{dx}+\dfrac{c^2}{x}=0}\;\;\;\mathsf{(\because\;xy=c^2\;\implies\;y=\dfrac{c^2}{x})}$

$\textsf{Multiply bothsides by 'x', we get}$

$\boxed{\bf\;\;x^2\;\dfrac{dy}{dx}+c^2=0}$

$\underline{\textbf{Product rule:}}$

$\boxed{\begin{minipage}{5cm} \\\mathsf{\dfrac{d(uv)}{dx}=u\;\dfrac{dv}{dx}+v\;\dfrac{du}{dx}} \\\end{minipage}}$