Maharashtra State BoardHSC Arts 12th Board Exam
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Solve the following differential equation: dydxy cot xxcot x2xdydx+y cot x=x2cot x+2x - Mathematics and Statistics

Sum

Solve the following differential equation:

`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`

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Solution

`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`  ....(1)

This is the linear differential equation of the form

`"dy"/"dx" + "Py" = "Q",` where P = cot x and Q = x2 cot x + 2x

∴ I.F. = `"e"^(int "P dx") = "e"^(int "cot x"  "dx")`

`= "e"^(log ("sin x")) = sin "x"`.

∴ the solution of (1) is given by

`"y" * ("I.F.") = int "Q" * ("I.F.")"dx" + "c"`

∴ y sin x = `int ("x"^2 "cot x" + "2x") "sin x dx" + "c"`

∴ y sin x = `int ("x"^2 cot "x" * sin "x" + "2x" sin "x") "dx" + "c"` 

∴ y sin x = `int "x"^2 cos "x"  "dx" + 2 int "x" sin "x"  "dx" + "c"`

∴ y sin x = `"x"^2 int "cos x  dx" - int["d"/"dx" ("x"^2) int "cos x  dx"] "dx" + 2 int "x sin x dx" + "c"`

`= "x"^2 (sin "x") - int "2x" (sin "x")"dx" + 2 int "x sin x dx" + "c"`

`= "x"^2 sin x - 2 int "x" sin "x"  "dx" + 2 int "x sin x dx" + "c"`

∴ y sin x = x2 sin x + c

∴ y = x2 + c cosec x 

This is the general solution.

Concept: Formation of Differential Equations
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 6 Differential Equations
Miscellaneous exercise 6 | Q 5.5 | Page 217
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