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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.

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

The given differential equation is

log space dy over dx space equals space ax plus by space space space space or space space space space dy over dx space equals space straight e to the power of ax plus by end exponent

dy over dx space equals space straight e to the power of ax. space straight e to the power of by.

Separating the variables, we get,

1 over straight e to the power of by dy space equals space straight e to the power of ax dx

Integrating,

integral straight e to the power of negative by end exponent dy space equals space straight e to the power of ax dx

therefore space space space space space space fraction numerator straight e to the power of negative by end exponent over denominator negative straight b end fraction space equals space straight e to the power of ax over straight a plus straight c space space space space space or space space space space minus 1 over straight b straight e to the power of negative by end exponent space equals space 1 over straight a straight e to the power of ax plus straight c

which is required solution.

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140.

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

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Previous Year Papers

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Short Answer Type

Long Answer Type

Short Answer Type

139. Solve:

139.
Solve:
$log dy over dx equals ax plus by.$