y ' = 3y(1 - (y/4)) , y(0) = 2
please show steps. thanks
please show steps. thanks
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dy/dx = 3y - 3y^2/4
dy = (3y - 3y^2/4) dx
dx = dy/(3y - 3y^2/4)
dx = 4 dy / (12y - 3y^2)
1/4 dx = dy / (3 * (4y - 3y^2))
3/4 dx = dy / (4y - y^2)
-3/4 dx = dy/(y^2 - 4y)
-3/4 dx = dy/[y*(y - 4)]
use partial fractions
A/y + B/(y - 4) = 1/[(y)(y - 4)]
A*(y - 4) + B*(y) = 1
y = 0
-4A = 1
A = -1/4
y = 4
4B = 1
B = 1/4
-3/4 dx = 1/(4*(y - 4)) - 1/(4y) dy
multiply everything out by four
-3 dx = 1/(y - 4) - 1/y dy
3 dx = 1/y - 1/(y - 4) dy
3 dx = 1/y + 1/(4 - y) dy
3x + C = ln(y) - ln(4 - y)
3x + C = ln[y/(4 - y)]
e^(3x + C) = y/(4 - y)
(4 - y) * e^(3x + C) = y
4 * e^(3x + C) - y * e^(3x + C) = y
4 * e^(3x + C) = y + y * e^(3x + C)
4 * e^(3x + C) = y * (e^(3x + C) + 1)
y = 4 * e^(3x + C) / [1 + e^(3x + C)]
y(x = 0) = 4 * e^(3x + C) / [1 + e^(3x + C)] = 2
4 * e^(3*0 + C) / [1 + e^(3*0 + C)] = 2
4 * e^C / (1 + e^C) = 2
4 * e^C = 2 + 2 * e^C
2 * e^C = 2
e^C = 1
C = 0
Final answer is
4 * e^(3x) / [1 + e^(3x)]
dy = (3y - 3y^2/4) dx
dx = dy/(3y - 3y^2/4)
dx = 4 dy / (12y - 3y^2)
1/4 dx = dy / (3 * (4y - 3y^2))
3/4 dx = dy / (4y - y^2)
-3/4 dx = dy/(y^2 - 4y)
-3/4 dx = dy/[y*(y - 4)]
use partial fractions
A/y + B/(y - 4) = 1/[(y)(y - 4)]
A*(y - 4) + B*(y) = 1
y = 0
-4A = 1
A = -1/4
y = 4
4B = 1
B = 1/4
-3/4 dx = 1/(4*(y - 4)) - 1/(4y) dy
multiply everything out by four
-3 dx = 1/(y - 4) - 1/y dy
3 dx = 1/y - 1/(y - 4) dy
3 dx = 1/y + 1/(4 - y) dy
3x + C = ln(y) - ln(4 - y)
3x + C = ln[y/(4 - y)]
e^(3x + C) = y/(4 - y)
(4 - y) * e^(3x + C) = y
4 * e^(3x + C) - y * e^(3x + C) = y
4 * e^(3x + C) = y + y * e^(3x + C)
4 * e^(3x + C) = y * (e^(3x + C) + 1)
y = 4 * e^(3x + C) / [1 + e^(3x + C)]
y(x = 0) = 4 * e^(3x + C) / [1 + e^(3x + C)] = 2
4 * e^(3*0 + C) / [1 + e^(3*0 + C)] = 2
4 * e^C / (1 + e^C) = 2
4 * e^C = 2 + 2 * e^C
2 * e^C = 2
e^C = 1
C = 0
Final answer is
4 * e^(3x) / [1 + e^(3x)]