Question

Form the differential equation by eliminating the arbitrary constant from the equation y= x^2-cx​

Answer 1

$\textbf{Given:}$

$\mathsf{y=x^2-c\,x}$

$\textbf{To find:}$

$\textsf{A differential equation by eliminating arbitrary constant c}$

$\textbf{Solution:}$

$\textsf{Differential equations are formed by eliminating}$

$\textsf{arbitrary constants in the equation}$

$\textsf{Consider,}$

$\mathsf{y=x^2-c\,x}$----------(1)

$\textsf{Differentiate with respect to x}$

$\mathsf{\dfrac{dy}{dx}=2x-c}$

$\mathsf{\dfrac{dy}{dx}-2x=c}$---------(2)

$\mathsf{Using (2) in (1), we get}$

$\mathsf{y=x^2-\left(\dfrac{dy}{dx}-2x\right)x}$

$\mathsf{y=x^2-x\,\dfrac{dy}{dx}+2x^2}$

$\implies\boxed{\mathsf{y=3x^2-x\,\dfrac{dy}{dx}}}$

$\textsf{which is the required differential equation}$

Answer 2

$\huge\textbf{Given:}$

$\mathsf{y=x^2-c\,x}$

$\huge\textbf{To find:}$

$\textsf{A differential equation by eliminating arbitrary constant c}$

$\huge\textbf{Solution:}$

$\textsf{Differential equations are formed by eliminating}$

$\textsf{arbitrary constants in the equation}$

$\textsf{Consider,}$

$\mathsf{y=x^2-c\,x}$----------(1)

$\textsf{Differentiate with respect to x}$

$\mathsf{\dfrac{dy}{dx}=2x-c}$

$\mathsf{\dfrac{dy}{dx}-2x=c}$---------(2)

$\mathsf{Using (2) in (1), we get}$

$\mathsf{y=x^2-\left(\dfrac{dy}{dx}-2x\right)x}$

$\mathsf{y=x^2-x\,\dfrac{dy}{dx}+2x^2}$

$\implies\boxed{\mathsf\red{y=3x^2-x\,\dfrac{dy}{dx}}}$