1. ## Differential Equations (2)

Solve the following differential equations using the substitution given.

$\displaystyle{\frac{dy}{dx}} + y = y^2$, by substituting z = 1/y, given that y = 0.5 when x =0.

Please guide me as to how to use the substitution....I'll try to figure out the rest on my own...Thank you!

2. Hi

This time, I won't make any confusion

$z=\frac 1y \implies y=\frac 1z$
$y(0)=0.5 \implies z(0)=2$

$\frac{dy}{dx}=\frac{d(\tfrac 1z)}{dx}=-\frac{\tfrac{dz}{dx}}{z^2}=-\frac{dz}{dx} \frac 1{z^2}$

Substitute $\frac{dy}{dx}$ and $y$ in the DE and multiply by $z^2$