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

The given differential equation is

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

Put

4 straight x plus straight y plus 1 space equals space straight t comma

therefore space space 4 plus dy over dx space equals space dt over dx comma space space or space space dy over dx space equals space dt over dx minus 4

therefore space space space given space equation space becomes comma space dt over dx minus 4 space equals straight t squared

dt over dx space equals space straight t squared plus 4

Separating the variables, we get,

fraction numerator dt over denominator straight t squared plus 4 end fraction space equals space dx

Integrating both sides,

integral fraction numerator dt over denominator straight t squared plus left parenthesis 2 right parenthesis squared end fraction space equals space integral 1. space dx

1 half tan to the power of negative 1 end exponent straight t over 2 space equals space straight x plus straight c space space space or space space tan to the power of negative 1 end exponent straight t over 2 space equals space 2 space left parenthesis straight x plus straight c right parenthesis comma space space or space space straight t over 2 space equals space tan space open square brackets 2 space left parenthesis straight x plus straight c right parenthesis close square brackets

therefore

straight t space equals space 2 space tan space left square bracket space 2 space left parenthesis straight x space plus straight c right parenthesis right square bracket

4 straight x plus straight y plus 1 space equals space 2 space tan space left square bracket 2 space left parenthesis straight x plus straight c right parenthesis right square bracket comma space which space is space the space required space solution.

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