Trying to solve this seemingly simple first order non-linear differential equation: y

Isaiah Farrell 2022-05-25 Answered
Trying to solve this seemingly simple first order non-linear differential equation:
y + y 2 = cos 2 x
Considered separation of variables and bernoulli methods but figured it's not applicable. Please I need a hint.
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Answers (1)

Ronnie Glenn
Answered 2022-05-26 Author has 11 answers
y + y 2 = cos ( 2 x )
Change of function: y = Y Y y = Y Y Y 2 Y 2
Y Y = cos ( 2 x )
The general solution involves the Mathieu functions:
Y = c 1 MathieuC ( 0 , 1 2 , x ) + c 2 MatieuS ( 0 , 1 2 , x )
y ( x ) = c 1 d d x MatieuC ( 0 , 1 2 , x ) + c 2 d d x MatieuS ( 0 , 1 2 , x ) c 1 MatieuC ( 0 , 1 2 , x ) + c 2 MatieuS ( 0 , 1 2 , x )

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