What is a solution to the differential equation dy/dx=2y−1?

Joyce Sharp 2022-09-23 Answered
What is a solution to the differential equation $\frac{dy}{dx}=2y-1$?
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Answers (1)

Zackary Galloway
Answered 2022-09-24 Author has 17 answers
Substituting $z=2y-1$ in $\frac{dy}{dx}=2y-1$
and considering
$\frac{dz}{dx}=2\frac{dy}{dx}$ we have
$\frac{1}{2}\frac{dz}{dx}=z$. Grouping variables
$\frac{dz}{z}=2dx$ and integrating
${\mathrm{log}}_{e}z=2x+{C}_{1}$ or equivalently
$z={C}_{2}{e}^{2x}=2y-1$ and finally
$y=\frac{{C}_{2}{e}^{2x}+1}{2}$
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