Question

The general solution of the differential equation 9yy'+4x = 0 is

Answer 1

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

$\mathsf{9\,y\,y'+4x=0}$

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

$\textsf{The general solution of the differential equation}$

$\mathsf{9\,y\,y'+4x=0}$

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

$\textsf{We apply Variable separable method to find the}$

$\textsf{solution of the equation}$

$\mathsf{Consider,}$

$\mathsf{9\,y\,y'+4x=0}$

$\textsf{This can be written as}$

$\mathsf{9\,y\,\dfrac{dy}{dx}+4x=0}$

$\mathsf{9\,y\,\dfrac{dy}{dx}=-4x}$

$\mathsf{9\,y\,dy=-4x\,dx}$

$\textsf{Integrating on bothsides, we get}$

$\mathsf{\int\,9\,y\,dy=\int(-4x)\,dx}$

$\mathsf{9\int\,y\,dy=-4\int\,x\,dx}$

$\textsf{Using the formula}$

$\boxed{\mathsf{\int\,x^n\,dx=\dfrac{x^{n+1}}{n+1}+C}}$

$\mathsf{9\left(\dfrac{y^2}{2}\right)=-4\left(\dfrac{x^2}{2}\right)+C}$

$\mathsf{\dfrac{9y^2}{2}=-2x^2+C}$

$\mathsf{9y^2=-4x^2+2C}$

$\bf\,4x^2+9y^2=K\;\;\;\;\;\textsf{(Where K=2C)}$

$\underline{\textbf{Answer:}}$

$\textsf{Solution of the differential equation is}$

$\bf\,4x^2+9y^2=K$