Solve the given differential equation. (x - a) (x - b) y' - (y -

c. = 0, where a, b, c are constants. (assume a ≠

b.)

(x - a) (x - b) y' - (y - c) = 0

Rewriting the equation we have

(x - a) (x - b) y' = (y - c)

(x - a) (x - b) (dy/dx) = (y - c)

separating the variables

dy / (y - c) = dx / ((x - a) (x - b))

integrating both sides of the equation

Left side:

Ln (y-c)

Right side:

Simple fractions

(1 / ((x - a) (x - b))) = (A / (x - a)) + (B / (x - b))

A = (1 / (a-b))

B = (1 / (b-a))

I[dx / ((x - a) (x - b)) ] = I[ ((1 / (a-b)) dx / (x - a)) ] + I[ ((1 / (b-a)) dx / (x - b)) ]

I[dx / ((x - a) (x - b)) ] = (1 / (a-b)) * Ln (x-a) + (1 / (b-a)) * Ln (x-b) + C

Then the solution is

Ln (y-c) = (1 / (a-b)) * Ln (x-a) + (1 / (b-a)) * Ln (x-b) + C

Rewriting

Exp[Ln (y-c) ] = exp[ (1 / (a-b)) * Ln (x-a) + (1 / (b-a)) * Ln (x-b) + C]

y-c = exp[ (1 / (a-b)) * Ln (x-a) ]*exp[ (1 / (b-a)) * Ln (x-b) ]*exp[C]

y-c=[ (x-a) ^ (1 / (a-b)) ]*[ (x-b) ^ (1 / (b-a)) ]*C’

final answer:

y=C’*[ (x-a) ^ (1 / (a-b)) ]*[ (x-b) ^ (1 / (b-a)) ] * + c