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181.

Solve:
$cos space left parenthesis straight x plus straight y right parenthesis space dy over dx space equals space 1.$

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182.

Solve:
$dy over dx space equals space tan space left parenthesis straight x plus straight y right parenthesis$

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183. Solve the following initial value problem:
(x + y + 1)² dy = dx, y ( –1) = 0

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184.

Solve the following initial value problem
cos (x + y) dy = dx, y (0) = 0

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185. For the following differential equation, given below, find particular solution satisfying the given condition:

open parentheses straight x cubed plus straight x squared plus straight x plus 1 close parentheses dy over dx space equals space 2 straight x squared plus straight x semicolon space space straight y space equals space 1 space space when space straight x space equals space 0

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186. For the following differential equation, given below, find particular solution satisfying the given condition:

straight x open parentheses straight x squared minus 1 close parentheses space dy over dx space equals space 1 space semicolon space space straight y space equals space 0 space space when space straight x space equals space 2

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187. For the following differential equation, given below, find particular solution satisfying the given condition:

cos space open parentheses dy over dx close parentheses space space equals straight a space space space left parenthesis straight a space element of space straight R right parenthesis semicolon space space space straight y space equals space 1 space space when space straight x space equals space 0

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The given differential equation is

dy over dx space equals space straight y space tanx

Separating the variables, we get,

1 over straight y dy space equals space tanx space dx

Integrating,

integral 1 over straight y dy space equals space integral tanx space dx

therefore space space space space log space open vertical bar straight y close vertical bar space equals space log space open vertical bar cosx close vertical bar space plus space straight c apostrophe space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis therefore space space space space log space open vertical bar straight y close vertical bar plus log space open vertical bar cosx close vertical bar space equals space straight c apostrophe therefore space space space log space open vertical bar straight y space cosx close vertical bar space equals space straight c apostrophe therefore space space space space open vertical bar straight y space cosx close vertical bar space equals space straight e to the power of straight c apostrophe end exponent therefore space space space straight y space cosx space equals space plus-or-minus straight e to the power of straight c space space space or space space straight y space cosx space equals space straight c

Now y = 1 when x = 0

therefore space space space 1 space space cos space 0 space equals space straight c space space space space rightwards double arrow space space space space 1 cross times 1 space equals space straight c space space space space space rightwards double arrow space space space straight c space equals space 1 therefore space space from space left parenthesis 1 right parenthesis comma space space space straight y space cosx space equals space 1 comma space space which space is space required space solution.

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189.

Solve: $straight x squared dy over dx space equals straight x squared plus 5 xy plus 4 straight y squared.$

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190.

Solve the following differential equation:
$straight x squared dy over dx space equals space 2 xy plus straight y squared$

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