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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

e^x + e^{– y} = C

The given differential equation is

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

Separating the variables, we get,

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

Integrating,

integral straight e to the power of negative straight y end exponent dy space equals space integral straight e to the power of straight x space dx

therefore space space space space fraction numerator straight e to the power of negative straight y end exponent over denominator negative 1 end fraction space equals space straight e to the power of straight x plus straight c apostrophe therefore space space space space space space minus straight e to the power of negative straight y end exponent space equals space straight e to the power of straight x plus straight c apostrophe space space or space space straight e to the power of straight x plus straight e to the power of negative straight y end exponent space equals space minus straight c apostrophe therefore space space space space straight e to the power of straight x plus straight e to the power of negative straight y end exponent space equals space straight C comma space which space is space required space solution. space therefore space space left parenthesis straight A right parenthesis space is space correct space answer.

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