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महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

Reduce the following differential equation to the variable separable form and hence solve: x - ydydxa(x - y)2dydx=a2

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प्रश्न

Reduce the following differential equation to the variable separable form and hence solve:

`("x - y")^2 "dy"/"dx" = "a"^2`

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उत्तर

`("x - y")^2 "dy"/"dx" = "a"^2`     .....(1)

Put x - y = u

∴ x - u = y

∴ 1 - `"du"/"dx" = "dy"/"dx"`

∴ (1) becomes, `"u"^2 (1 - "du"/"dx") = "a"^2`

∴ `"u"^2 - "u"^2 "du"/"dx" = "a"^2`

∴ `"u"^2 - "a"^2 = "u"^2 "du"/"dx"`

∴ dx = `"u"^2/("u"^2 - "a"^2)`du

Integrating both sides, we get

`int "dx" = int (("u"^2 - "a"^2) + "a"^2)/("u"^2 - "a"^2)`du

∴ x = `int 1  "du" + "a"^2 int "du"/("u"^2 - "a"^2) + "c"_1`

`1/"2a" log |("u - a")/("u + a")| + "c"_1`

∴ x = x - y + `"a"/2 log |("x - y - a")/("x - y + a")| + "c"_1`

∴ - c1 + y = `"a"/2 log |("x - y - a")/("x - y + a")|`

∴ - 2c1 + 2y =  a log `|("x - y - a")/("x - y + a")|`

∴ c + 2y = a log `|("x - y - a")/("x - y + a")|`, where c = - 2c1

This is the general solution.

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पाठ 6: Differential Equations - Exercise 6.3 [पृष्ठ २०१]

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बालभारती Mathematics and Statistics 2 (Arts and Science) [English] Standard 12 Maharashtra State Board
पाठ 6 Differential Equations
Exercise 6.3 | Q 4.2 | पृष्ठ २०१

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