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

The given differential equation is

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

...(1)
Put

straight x plus straight y space equals straight t comma space space space space space space space space space space space therefore space space space space 1 plus dy over dx equals space dt over dx space space or space space space dy over dx space equals dt over dx minus 1

therefore space space space from space left parenthesis 1 right parenthesis comma space space dt over dx minus 1 plus 1 space equals space straight e to the power of straight t space space space space space or space space space dt over dx space equals straight e to the power of straight t rightwards double arrow space space space space 1 over straight e to the power of straight t dt space equals space dx space space space rightwards double arrow space space space integral straight e to the power of negative straight t end exponent space dt space equals space integral 1. space dx therefore space space space space space space space space space space fraction numerator straight e to the power of negative straight t end exponent over denominator negative 1 end fraction space equals space straight x plus straight c space space space space rightwards double arrow space space space minus straight e to the power of negative left parenthesis straight x plus straight y right parenthesis end exponent space equals space straight x space plus straight c space space which space is space required space solution. space space space

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

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