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n-o-oo-1-n-x-2n-1-4n-2-1-

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Question Number 184858 by SANOGO last updated on 12/Jan/23
Σ_(n=o) ^(+oo) (((−1)^n x^(2n+1) )/(4n^2 −1))
∑+oon=o(−1)nx2n+14n2−1
Answered by qaz last updated on 14/Jan/23
y=Σ(((−1)^n )/(4n^2 −1)) x^(2n+1) y′=Σ(((−1)^n x^(2n) )/(2n−1))=xΣ(((−1)^n x^(2n−1) )/(2n−1)) y′′=Σ(((−1)^n x^(2n−1) )/(2n−1))+xΣ(−1)^n x^(2n−2) =((y′)/x)+(1/(x(1+x^2 ))) ⇒y′=C_1 x−1−xarctan x ⇒y=C_1 x^2 −(1/2)x−(1/2)x^2 arctan x−(1/2)arctan x+C_2 y(0)=0 y(0)′=−1 ⇒C_1 =0 C_2 =0 ⇒y=−(1/2)x−(1/2)x^2 arctan x−(1/2)arctan x
y=Σ(−1)n4n2−1x2n+1y′=Σ(−1)nx2n2n−1=xΣ(−1)nx2n−12n−1y″=Σ(−1)nx2n−12n−1+xΣ(−1)nx2n−2=y′x+1x(1+x2)⇒y′=C1x−1−xarctanx⇒y=C1x2−12x−12x2arctanx−12arctanx+C2y(0)=0y(0)′=−1⇒C1=0C2=0⇒y=−12x−12x2arctanx−12arctanx
Commented by SANOGO last updated on 14/Jan/23
thank you
thankyou

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