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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

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

The given differential equation is

dy over dx space equals space sin space 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 space dy over dx space equals space dt over dx minus 1

therefore space space space space space space from space left parenthesis 1 right parenthesis comma space space dt over dx minus 1 space equals space space sin space straight t space space space or space space dt over dx space equals space 1 plus sin space straight t

Separating the variables, we get,

fraction numerator 1 over denominator 1 plus sin space straight t end fraction dt space equals space dx

Integrating both sides, we get,

integral fraction numerator 1 over denominator 1 plus sin space straight t end fraction dt space equals space integral 1. space dx space space space space or space space space space integral fraction numerator 1 over denominator 1 plus sin space straight t end fraction cross times fraction numerator 1 minus sin space straight t over denominator 1 minus sin space straight t end fraction dt space equals space integral 1. space dx

integral fraction numerator 1 minus sint over denominator 1 minus sin squared straight t end fraction dt space equals space integral 1. space dx space space space space space space space space space space space or space space space space space fraction numerator 1 minus sin space straight t over denominator cos squared straight t end fraction dt space equals space integral 1. space dx

integral open parentheses fraction numerator 1 over denominator cos squared straight t end fraction minus fraction numerator sin space straight t over denominator cos squared straight t end fraction close parentheses dt space equals space integral 1. space dx space space or space space integral open parentheses fraction numerator 1 over denominator cos squared straight t end fraction minus fraction numerator 1 over denominator cos space straight t end fraction. fraction numerator sin space straight t over denominator cos space straight t end fraction close parentheses dt space equals space integral 1. space dx

open parentheses sec squared straight t minus sect space tant close parentheses space dt space equals space 1. space dx

tant space minus sect space equals space straight x plus straight c

tan left parenthesis straight x plus straight y right parenthesis space minus space sec left parenthesis straight x plus straight y right parenthesis space equals space straight x plus straight c

which is required solution.

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179.

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

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180.

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

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