Question

Find the particular solution for the following differential equation:

`sqrt(1 − y^2) dx = (sin^(−1) y − x)dy,` given that y(0) = 0

Answer 1

Standard Linear Form

Rearrange as a linear equation in x:

`dx/dy + x/(sqrt(1 − y^2)) = sin^(−1)y/(sqrt1 − y^2)`

I.F. = `""_e^∫ 1/sqrt(1 − y^2) dy = e sin^(−1)y`   ...[Integrating Factor (I.F.)]

= x × (I.F.) = ∫ Q(y).(I.F.)dy + C

`xesin^(−1) = ∫ (sin^(−1) y.e^(sin^(−1)y))/sqrt(1 − y^2) dy + C`

= ∫ te^t dt = e^t(t − 1)   ...[Substituting t = sin⁻¹ y, the integral becomes] 

`xe^(sin^(−1)y) = e^(sin^(−1)y)(sin^(−1) y − 1) + C`

Apply y(0) = 0 (meaning x = 0 when y = 0):

0 = e⁰(0 − 1) + C ⇒ C = 1

x = `sin^(−1) y − 1 + e^(−sin^(−1)y)`