What is a solution to the differential equation $\frac{d y}{d x} = \frac{3 y}{2 + x}$ $dy/dx=(3y)/(2+x)$ ?

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Answer 1 · 2016-07-19T20:02:57

$y = e^{3 C} {(2 + x)}^{3}$ $y = e^(3C)(2+x)^(3)$

$= C {(2 + x)}^{3}$ $= C(2+x)^(3)$

This differential equation is separable.

$\frac{d y}{d x} = \frac{3 y}{2 + x}$ $dy/(dx) = (3y)/(2+x)$

$d y = \frac{3 y}{2 + x} d x$ $dy = (3y)/(2+x) dx$

$\frac{1}{3 y} d y = \frac{1}{2 + x} d x$ $1/(3y) dy = 1/(2+x) dx$

$\int \frac{1}{3 y} d y = \int \frac{1}{2 + x} d x$ $int 1/(3y) dy = int 1/(2+x) dx$

$\frac{1}{3} \int \frac{1}{y} d y = \int \frac{1}{2 + x} d x$ $1/3 int 1/y dy = int 1/(2+x) dx$

$\frac{1}{3} ln y = ln (2 + x) + C$ $1/3 ln y = ln(2+x) + C$

$ln y = 3 [ln (2 + x) + C]$ $ln y = 3[ln(2+x)+C]$

$y = e^{3 [ln (2 + x) + C]}$ $y = e^(3[ln(2+x)+C])$

$y = e^{3 C} {(2 + x)}^{3}$ $y = e^(3C)(2+x)^(3)$

$y = C {(2 + x)}^{3}$ $y = color(red)(C)(2+x)^(3)$

[where C is generic constant]

What is a solution to the differential equation dy/dx=(3y)/(2+x)dydx=3y2+xdy/dx=(3y)/(2+x)?