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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

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179.
Solve: sin (x+y) $dy over dx space equals 1$ .

The given differential equation is

sin space left parenthesis straight x plus straight y right parenthesis space dy over dx space equals space 1 space space space space space or space space dy over dx space equals space fraction numerator 1 over denominator sin space left parenthesis straight x plus straight y right parenthesis end fraction

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

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

therefore space space space space space from space left parenthesis 1 right parenthesis comma space dt over dx minus 1 space equals space fraction numerator 1 over denominator sin space straight t end fraction space space rightwards double arrow space space space dt over dx space equals space 1 plus fraction numerator 1 over denominator sin space straight t end fraction therefore space space space space dt over dx space equals space fraction numerator 1 plus sint over denominator sint end fraction space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space fraction numerator sint over denominator 1 plus sint end fraction dt space equals space dx therefore space space space space integral fraction numerator sin space straight t over denominator 1 plus sin space straight t end fraction dt space equals space integral 1. space dx space space space space rightwards double arrow space space space space integral fraction numerator left parenthesis 1 plus sin space straight t right parenthesis minus 1 over denominator 1 plus sint end fraction dt space equals space integral 1. space dx therefore space space space space integral space open parentheses 1 minus fraction numerator 1 over denominator 1 plus sin space straight t end fraction close parentheses dt space equals space integral 1. space dx rightwards double arrow space space integral open parentheses 1 minus 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 close parentheses space dt space equals space integral space 1. space dx rightwards double arrow space space integral open parentheses 1 minus fraction numerator 1 minus sin space straight t over denominator 1 minus sin squared straight t end fraction close parentheses space dt space equals space integral 1 space dx space space space space space space space space space rightwards double arrow space space space space space space space integral open parentheses 1 minus fraction numerator 1 minus sin space straight t over denominator cos squared straight t end fraction close parentheses dt space equals space integral 1. dx rightwards double arrow space space space integral open parentheses 1 minus fraction numerator 1 over denominator cos squared straight t end fraction plus 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 space dt space equals space integral space 1 space. space dx rightwards double arrow space integral left parenthesis 1 minus sec squared straight t space plus space sect space tant right parenthesis space dt space equals space integral 1. space dx rightwards double arrow space space space straight t minus tan space straight t space plus sec space straight t space equals space straight x space plus straight c therefore space space space straight x plus straight y minus tan space left parenthesis straight x plus straight y right parenthesis plus sec left parenthesis straight x plus straight y right parenthesis space equals space straight x plus straight c therefore space space straight y minus tan left parenthesis straight x plus straight y right parenthesis plus sec left parenthesis straight x plus straight y right parenthesis space equals space straight c space is space the space required space solution.

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180.

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