Question

Find the general/particular solution of the following Differential Equations

(Exact D.E)

4.) (dx/y) - x (dy/y^2)=0

Answer 1

The equation that we have to solve is:

$\frac{1}{y}{dx}-\frac{x}{y^2}{dy}=0 \iff M{dx}+N{dy}=0$

Then we confirm this is an exact differential equation when we confirm that $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$:

$\\M=\dfrac{1}{y};N=-\dfrac{x}{y^2};\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}=-\dfrac{1}{y^2}$

Then we proceed to solve for F(x,y):

\\ F(x,y)=\int M dx=\dfrac{1}{y}\int {dx}=\dfrac{x}{y}+g(y) \\ \frac{\partial F(x,y)}{\partial y}=-\dfrac{x}{y^2}+g\,'(y)=N \\ N=-\dfrac{x}{y^2} \implies g\,'(y)=0 \iff g(y)=C=constant \\ \implies \text{in conclusion } F(x,y)=\dfrac{x}{y}+C