Question

Obtain the differential equation by eliminating the arbitrary constants from the given equation, $y^{2}=(x+c)^{3}$

Answer 1

To obtain the differential equation by eliminating the arbitrary constants from the given equation, $y^{2}=(x+c)^{3}$

we had to differentiate given eq once and eliminate the constant as shown

$y^{2}=(x+c)^{3}...eq1 \\ \\ 2y \: \frac{dy}{dx} = 3( {x + c)}^{2} ...eq2 \\ \\ from \: eq1 \\ \\ (x + c) = {y}^{ \frac{2}{3} } \\ \\ so \: on \: squaring \: both \: sides \: we \: get \: \\ \\ {(x + c)}^{2} = {y}^{ \frac{4}{3} }$
place this value in eq 2,as shown below

$2y \: \frac{dy}{dx} = 3( {y}^{ \frac{4}{3} } ) \\ \\2 \frac{dy}{dx} = 3 \times \frac{{y}^{ \frac{4}{3} }}{y} \\ \\ \\2 \frac{dy}{dx} = 3 \times {y}^{ \frac{4 - 3}{3} } \\ \\ 2 \frac{dy}{dx} = 3 \times {y}^{ \frac{1}{3} } \\ \\ or \\ \\ 2 \frac{dy}{dx} - 3 {y}^{ \frac{1}{3} } = 0$
is the required differential equation.