Linearizing y′′+\sin(2x+\cos(2y′+y))+1- \sin(y+3y′)=0

Mary Buchanan 2022-01-20 Answered
Linearizing y+sin(2x+cos(2y+y))+1sin(y+3y)=0
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Tiefdruckot
Answered 2022-01-20 Author has 46 answers
The easiest linearization I can think of in this case is to use cos(θ)1,and sin(θ)θ where θ is small enough. In your case, these assumptions boil down to making the assumption that y, y0.
With these assumptions, your differential equation now becomes y+sin(2x+1)+1(y+3y)=0
which in turn becomes
y3yy=(1+sin(2x+1))
Now you can use your differential equation tricks to solve the above equation to get
y(x)=c1exp((3132)x)+c2exp
((3+132)x)+5sin(2x+1)6cos(2x+1)+6161
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