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: y'=cosx + y^2 >
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dm1dm
, y=y(x) y'=cosx + y^2, y(0)=1
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Dimka
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dm1dm

y'-y^2=0
y'=y^2
dy/dx=y^2
dy/y^2=dx
-1/y=x+C
y=-1/(x+C)

y1=Acosx+Bsinx
y1'=-Asinx+Bcosx
-Asinx+Bcosx-A(cosx)^2-B(sinx)^2=cosx
B=1

y=-1/(x+)+cosx

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dm1dm
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dm1dm
, y=y(x)
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dm1dm
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0+1(-)+2(-)^2+...+an(x-a)^n+...
-
. a_k . ?
dm1dm
y(0)=1, =1, - ?
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,
a_k=f^(k)(0)/(k!),
f^(k)(0) - k- x=0.
dm1dm

y'=cosx+y^2
y''=-sinx +2yy'
y'''=-cosx+2yy''+2(y')^2
y''''=sinx+2yy'''+6y'y''
y'''''=cosx+2yy''''+8y'y''''+6(y'')^2

y(0)=1
y'(0)=2
y''(0)=2
y'''(0)=11


y=1+2x/1!+2x^2/2!+...=1+2x+x^2

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-
, y'''(0)=3.
dm1dm
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