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Differential Equations

Short Answer Type

131. Solve 4 e^x tan y dx + 3(1 + e^x) sec²y dy = 0.

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Long Answer Type

132.

For the following differential equation, find the general solution:
sec² x tan y dx – sec² y tan x dy = 0.

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Short Answer Type

133.
For the following differential equation, find the general solution:
sec² x tan y dx + sec² y tan x dy = 0.

The given differential equation is
sec² x tan y dx + sec² y tan x dy = 0
or

sec squared straight y space tanx space dy space equals space sec squared straight x space tan space straight y space dx

Separating the variables, we get,

fraction numerator sec squared over denominator tan space straight y end fraction dy space equals space minus fraction numerator sec squared straight x over denominator tan space straight x end fraction dx

Integrating,

log space open vertical bar tan space straight y close vertical bar space equals space minus log space open vertical bar tan space straight x close vertical bar space plus space log space straight A

log space open vertical bar tan space straight y close vertical bar space plus space log space open vertical bar tan space straight x close vertical bar space equals space log space straight A space or space log space open vertical bar tan space straight x space tany close vertical bar space equals space log space straight A

open vertical bar tan space straight x space tan space straight y close vertical bar space equals space straight A space space space space space space rightwards double arrow space space space tan space straight x space tan space straight y space equals space plus-or-minus straight A space space space or space space space tanx space tany space equals space straight c

where c is an arbitrary constant.

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134.

Solve:
3e^x tan y dx + (1 – e^x) sec²y dy = 0.

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135.

Solve:
e^x tan y dx + (1 – e^x) sec² y dy = 0.

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136.

Solve:
tan y dx + sec² y tan x dy = 0.

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137.

Solve:
$left parenthesis 1 plus straight y squared right parenthesis thin space left parenthesis 1 plus log space straight x right parenthesis space dx space plus space straight x space dy space equals space 0$

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138.

Solve:
$left parenthesis 1 plus straight e to the power of 2 straight x end exponent right parenthesis space dy space plus space left parenthesis 1 plus straight y squared right parenthesis space straight e to the power of straight x space dx space equals space 0$

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139.

Solve:
$log dy over dx equals ax plus by.$

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140.

Solve:
$straight x squared left parenthesis straight y minus 1 right parenthesis space dx space plus space straight y squared space left parenthesis straight x minus 1 right parenthesis space dy space equals space 0$