Home » Blog » Find a first-order differential equation having the family y4 (x + 2) = C(x – 2) as integral curves

Find a first-order differential equation having the family y4 (x + 2) = C(x – 2) as integral curves

by RoRi

Find a first-order differential equation having the family

    \[ y^4 (x+2) = C(x-2) \]

as integral curves.

First, we differentiate both sides of the given equation with respect to x,

    \[ y^4 (x+2) = C(x-2) \quad \implies \quad 4y^3 y' (x+2) + y^4 = C. \]

From the original equation we can solve for the constant,

    \[ y^4 (x+2) = C(x-2) \quad \implies \quad C = \frac{y^4(x+2)}{x-2}. \]

Therefore we have,

    \begin{align*} &&4y^3 y' (x+2) + y^4 &= C \\ \implies && 4y^3 y' (x+2) + y^4 &= \frac{y^4(x+2)}{x-2} \\ \implies && 4y^3 y' (x^2-4) + y^4(x-2) &= y^4(x+2) \\ \implies && 4y^3 (x^2-4) y' - 4y^4 &= 0 \\ \implies && (x^2-4) y' - y &= 0. \end{align*}

This is a first-order differential equation with the given family of curves as integral curves.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):