solve-the-differential-equation-x-2-1-dy-dx-2y-0-when-y-3-and-x-2-expressing-your-answer-in-the-form-y-f-x-

Question Number 75175 by Rio Michael last updated on 08/Dec/19

s o l v e t h e d i f f e r e n t i a l e q u a t i o n

(x^{2} - 1) \frac{d y}{d x} + 2 y = 0 w h e n y = 3 a n d x = 2, e x p r e s s i n g

y o u r a n s w e r i n t h e f o r m y = f (x)

Answered by Kunal12588 last updated on 08/Dec/19

(x^{2} - 1) \frac{d y}{d x} + 2 y = 0

\Rightarrow \frac{d y}{2 y} = - \frac{d x}{x^{2} - 1}

\Rightarrow \frac{1}{2} \int \frac{d y}{y} = \int \frac{d x}{1 - x^{2}}

\frac{1}{1 - x^{2}} = \frac{A}{(1 - x)} + \frac{B}{(1 + x)}

\Rightarrow 1 = A (1 + x) + B (1 - x)

\Rightarrow A = \frac{1}{2}, B = \frac{1}{2}

\frac{1}{2} l o g (y) = \frac{1}{2} \int \frac{d x}{1 - x} + \frac{1}{2} \int \frac{d x}{1 + x}

\Rightarrow l o g y = - l o g (1 - x) + l o g (1 + x) + l o g C

\Rightarrow y = \frac{(1 + x) C}{1 - x}

3 = \frac{3 C}{- 1}

\Rightarrow C = - 1

y = \frac{x + 1}{x - 1}

Commented by Rio Michael last updated on 08/Dec/19

t h a n k y o u s i r