Solve: from Mathematics Differential Equations

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Differential Equations

Long Answer Type

171. The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20.000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009 ?

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Multiple Choice Questions

172. The general solution of the differential equation

dy over dx space equals space straight e to the power of straight x plus straight y end exponent

is

e^x + e^{– y} = C
e^x+ e^y = C
e^{– x} + e^y = C
e^{– x} + e^y = C

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Short Answer Type

173.

Solve $dy over dx space equals space left parenthesis 4 straight x plus straight y plus 1 right parenthesis squared.$

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Long Answer Type

174.

Solve $left parenthesis straight x minus straight y right parenthesis squared space dy over dx space equals space straight a squared.$

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Short Answer Type

175. Solve the following differential equation:

open parentheses straight x plus straight y close parentheses squared dy over dx space equals straight a squared

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176. Solve the following differential equation:

open parentheses straight x plus straight y plus 2 close parentheses space dy over dx space equals 2

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177. Solve the following differential equation:

dy over dx plus 1 space equals space straight e to the power of straight x plus straight y end exponent

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Long Answer Type

178. Solve the differential equation

dy over dx space equals sin left parenthesis straight x plus straight y right parenthesis space space or space sin to the power of negative 1 end exponent open parentheses dy over dx close parentheses space equals space straight x plus straight y

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179.

Solve: sin (x+y) $dy over dx space equals 1$ .

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180.
Solve:
$dy over dx space equals space cos space left parenthesis straight x plus straight y right parenthesis space space or space space cos to the power of negative 1 end exponent open parentheses dy over dx close parentheses space equals space straight x plus straight y$

The given differential equation is

dy over dx space equals cos left parenthesis straight x plus straight y right parenthesis

...(1)
Put x + y = t,

therefore space space space 1 plus dy over dx space equals space dt over dx space space or space space dy over dx space equals dt over dx minus 1

therefore space space from space left parenthesis 1 right parenthesis comma space space dt over dx minus 1 space equals space cos space straight t space space or space space dt over dx equals space 1 plus space cos space straight t therefore space space space space fraction numerator 1 over denominator 1 plus cos space straight t end fraction dt space space equals space dx space space space space rightwards double arrow space space space space integral fraction numerator 1 over denominator 1 plus cos space straight t end fraction dt space equals space integral 1. space dx rightwards double arrow space space space space integral fraction numerator 1 over denominator 1 plus cos space straight t end fraction cross times fraction numerator 1 minus cos space straight t over denominator 1 minus cos space straight t end fraction dt space equals space integral 1. space dx rightwards double arrow space space space space integral fraction numerator 1 minus cos space straight t over denominator 1 minus cos squared straight t end fraction dt space equals space integral 1. space dx space space space rightwards double arrow space space space space integral fraction numerator 1 minus cos space straight t over denominator sin squared straight t end fraction dt space equals space integral 1. space dx rightwards double arrow space integral open parentheses fraction numerator 1 over denominator sin squared straight t end fraction minus fraction numerator cos space straight t over denominator sin squared straight t end fraction close parentheses space dt space equals space integral 1. space dx space space space rightwards double arrow space space space space integral left parenthesis cosec squared straight t minus cosect space cot space straight t right parenthesis space dx space equals space integral 1. space dx rightwards double arrow space space space minus cot space straight t space plus space cosec space straight t space equals straight x plus straight c rightwards double arrow space space space space space fraction numerator cos space straight t over denominator sin space straight t end fraction plus fraction numerator 1 over denominator sin space straight t end fraction space equals space straight x plus straight c space space space space rightwards double arrow space space space space space fraction numerator 1 minus cos space straight t over denominator sin space straight t end fraction space equals space straight x plus straight c rightwards double arrow space space space fraction numerator 2 space sin squared begin display style straight t over 2 end style over denominator 2 space sin begin display style straight t over 2 end style space cos begin display style straight t over 2 end style end fraction space equals space straight x plus straight c space space space rightwards double arrow space space space tan straight t over 2 space equals space straight x plus straight c space space space rightwards double arrow space space space tan fraction numerator straight x plus straight y over denominator 2 end fraction space equals straight x plus straight c space space space which space is space required space solution. space

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