MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Elimination Method of Finding Solution of a Pair of Linear Equations

x x 2-1 dydx=...

Question

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

Open in App

Solution

$We have, x (x^{2} - 1) \frac{d y}{d x} = 1 \Rightarrow \frac{d y}{d x} = \frac{1}{x (x^{2} - 1)} \Rightarrow d y = \{\frac{1}{x (x^{2} - 1)}\} d x Integrating both sides, we get \int d y = \int \{\frac{1}{x (x^{2} - 1)}\} d x \Rightarrow y = \int \{\frac{1}{x (x^{2} - 1)}\} d x \Rightarrow y = \int \frac{1}{x (x - 1) (x + 1)} d x Let \frac{1}{x (x - 1) (x + 1)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x + 1} \Rightarrow 1 = A (x^{2} - 1) + B (x^{2} + x) + C (x^{2} - x) \Rightarrow 1 = (A + B + C) x^{2} + (B - C) x - A Equating the coefficients on both sides we get A + B + C = 0 . . . . . (1) B - C = 0 . . . . . (2) A = - 1 . . . . . (3) Solving (1), (2) and (3), we get A = - 1 B = \frac{1}{2} C = \frac{1}{2} ∴ y = \frac{1}{2} \int \frac{1}{x - 1} d x - \int \frac{1}{x} d x + \frac{1}{2} \int \frac{1}{x + 1} d x = \frac{1}{2} \log |x - 1| - \log |x| + \frac{1}{2} \log |x + 1| + C = \frac{1}{2} \log |x - 1| + \frac{1}{2} \log |x + 1| - \log |x| + C It is given that y (2) = 0 . ∴ 0 = \frac{1}{2} \log |2 - 1| + \frac{1}{2} \log |2 + 1| - \log |2| + C \Rightarrow C = \log |2| - \frac{1}{2} \log |3| Substituting the value of C, we get y = \frac{1}{2} \log |x - 1| + \frac{1}{2} \log |x + 1| - \log |x| + \log |2| - \frac{1}{2} \log |3| \Rightarrow 2 y = \log |x - 1| + \log |x + 1| - 2 \log |x| + 2 \log |2| - \log |3| \Rightarrow 2 y = \log |x - 1| + \log |x + 1| - \log |x^{2}| + \log 4 - \log 3 \Rightarrow 2 y = \log \frac{4 (x - 1) (x + 1)}{3 x^{2}} \Rightarrow y = \frac{1}{2} \log \frac{4 (x^{2} - 1)}{3 x^{2}} Hence, y = \frac{1}{2} \log \frac{4 (x^{2} - 1)}{3 x^{2}} is the solution to the given differential equation .$

Suggest Corrections

thumbs-up

0

Similar questions

Q. If y = cosec⁻¹ x, x >1, then show that

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

Q. For each of the following differential equations, find a particular solution satisfying the given condition:
(i)

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

\cos (\frac{d y}{d x}) = a, y = 1 when x = 0

\frac{d y}{d x} = y \tan x, y = 1 when x = 0

y \sqrt{x^{2} + 1} = \log (\sqrt{x^{2} + 1} - x)

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

x (x^{2} - 1) (x + 2) + 1 = 0

(x + y + 1) \frac{d y}{d x} = 1

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Algebraic Solution

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Elimination Method of Finding Solution of a Pair of Linear Equations

Standard X Mathematics

Join BYJU'S Learning Program

CrossIcon