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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.

The given differential equation is
sec² x tan y dx – sec² y tan x dy = 0.
or sec² x tan y dx = sec²y tan x dy
or

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

Integrating, we get,

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

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

or space space log space open vertical bar tan space straight y close vertical bar space minus space log space open vertical bar tan space straight x close vertical bar space equals space log space straight A or space space log space open parentheses fraction numerator open vertical bar tan space straight y close vertical bar over denominator open vertical bar tan space straight x close vertical bar end fraction close parentheses space equals space log space space straight A space space space space space space space space space space space space space space space space space space space space space space space space or space space space log space open vertical bar fraction numerator tan space straight y over denominator tan space straight x end fraction close vertical bar space equals space log space straight A or space space space space space space space space space space open vertical bar fraction numerator tan space straight y over denominator tan space straight x end fraction close vertical bar space equals straight A space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space or space space space space fraction numerator tan space straight y over denominator tan space straight x end fraction space equals space plus-or-minus space straight A or space space space space space fraction numerator tan space straight y over denominator tan space straight x end fraction space equals straight c comma space space space where space straight c space is space arbitrary space constant. space or space space space space space tan space straight y space equals space straight c space tanx space comma space which space is space required space solution. space

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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.

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