Find a solution of x dydx y2 y that passes through the ind

Find a solution of x dy/dx = y^2 ? y that passes through the indicated points. (a) (0, 1) y = ? (b) (0, 0) y = ? (c)1 4 , 1 4 y = ? (d) 4, 1 6 y = ?

Solution

x*dy/dx = y^2 - y

dy/(y^2 - y) = dx/x

1/(y-1) - 1/y = dx/x

Integrating :

ln|y - 1| - ln|y| = ln|x| + D

ln|(y - 1)/y| = ln|Cx|

1 - 1/y = Cx

1/y = 1 - Cx

y = 1 / (1 - Cx)

a)

(0 , 1) :

1 = 1 / (1 - C*0)
1 = 1/1
1 = 1
ALWAYS TRUE

So, c can be anything

Let c = 1...
y = 1 / (1 - x) --> ANSWER

b)

(0 , 0)

y = 1 / (1 - Cx)
0 = 1 / (1 - C*0)
0 = 1 / 1
0 = 1
ALWAYS FALSE

So, no solution exists

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c)

14 = 1 / (1 - 14C)

1 - 14C = 1/14

14C = 1 - 1/14

14C = 13/14

C = 13/196

So, y = 1 / (1 - 13x/196)

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d)

(4 , 16)

16 = 1 / (1 - 4C)

1 - 4C = 1/16

1 - 1/16 = 4C

4C = 15/16

C = 15/64

So, y = 1 / (1 - 15x/64)