Question

Find differential equation corresponding to $y=c(x-c{)}^2$, where $c$ is arbitrary constant.

• A. $y^{\prime 3}=4y(x{y}^{\prime }-2y)$
• B. $y^{\prime 3}=4y(x{y}^{\prime }+2y)$
• C. $y^{\prime 3}=-8y(x{y}^{\prime }-2y)$
• D. $y^{\prime 2}=-12y(x{y}^{\prime }-2y)$

Answer 1

The correct option is A $y^{\prime 3}=4y(x{y}^{\prime }-2y)$

$y=c(x-c{)}^2$

On differentiating, we get

$y^{{}^{\prime }}=\frac{dy}{dx}=2c(x-c)$

$(y^{{}^{\prime }}{)}^3=c^3(x-c{)}^3$

$y{y}^{{}^{\prime }}=2c^2(x-c{)}^3$

$\frac{y}{y^{{}^{\prime }}}=\frac{x-c}{2}$

$c=x-\frac{2y}{y^{{}^{\prime }}}$

Putting c in y ,we get

$\to y=(x-\frac{2y}{y^{{}^{\prime }}})(\frac{2y}{y^{{}^{\prime }}}{)}^2$

$\to (y^{{}^{\prime }}{)}^3=4y(x{y}^{{}^{\prime }}-2y)$