What is a general solution to the differential equation dy/dx=10-2y?

1 Answer
Jul 19, 2016

y = (e^(-2(x+C)) + 10)/(2)

OR

y = C e^(-2x) +5

Explanation:

This differential equation is separable.

dy/(dx) = 10-2y

dy = (10-2y) dx

1/(10-2y) dy = dx

int dy/(10-2y) = int dx

Let u = 10-2y -> du = -2 dy -> -1/2 du = dy

-1/2 int (du)/(u) = int x dx

-1/2 ln(10-2y) = x + C

ln(10-2y) = -2(x+C)

10-2y = e^(-2(x+C))

-2y+10 = e^(-2(x+C))

-2y = e^(-2(x+C)) - 10


color(red)(-2y = e^(-2x)e^(-2C) - 10)

color(red)(-2y = C e^(-2x) - 10)

color(red)(y = C/(-2) e^(-2x) - 10/(-2))

color(red)(y = C e^(-2x) +5)


y = - (e^(-2(x+C)) -10)/(2)

y = (e^(-2(x+C)) + 10)/(2)