Solve the differential equation (x^2 -1)dy/dx =-(x^2+1)y

Question

Given y=1 when x=3. Express y as a function of x

 

Answer 1

the equation becomes y' = -(x²+1)/(x²-1)y y(3) = 1

When you break down this fraction into all the partial fractions and integrate one at a time, you get

ln(y) = -x + ln(x+1) - ln(x-1) + constant

or

y = exp( -x + ln(x+1) - ln(x-1) + constant )

y(3) = exp( -3 + ln(3+1) - ln(3-1) + constant )

1 = exp( -2.307 + constant) so, constant = 2.307