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linear 4xy^{\prime \prime}+2y^{\prime}+y=0

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linear 4xy^{′ ′}+2y^′+y=0

Solution

y=c₁√1√x sin(√x)⁴√x+c₂√1√x cos(√x)⁴√x

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(x+4)⁴

Aplicar o teorema do binômio: (a+b)ⁿ=∑_i=0ⁿ(ni)a⁽ⁿ⁻ⁱ⁾bⁱ

a=x, b=4

=∑_i=0⁴(4i)x⁽⁴⁻ⁱ⁾· 4ⁱ

Expandir a soma

=4!0!(4−0)! x⁴· 4⁰+4!1!(4−1)! x³· 4¹+4!2!(4−2)! x²· 4²+4!3!(4−3)! x¹· 4³+4!4!(4−4)! x⁰· 4⁴

Simplificar 4!0!(4−0)! x⁴· 4⁰: x⁴

Simplificar 4!1!(4−1)! x³· 4¹: 16x³

Simplificar 4!2!(4−2)! x²· 4²: 96x²

Simplificar 4!3!(4−3)! x¹· 4³: 256x

Simplificar 4!4!(4−4)! x⁰· 4⁴: 256

=x⁴+16x³+96x²+256x+256

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−▭▭	<	7	8	9	÷	`AC`
+▭▭	>	4	5	6	×	☐☐☐
×▭▭	(	1	2	3	−	`x`
▭ \|▭	)	.	0	=	+	`y`

linear 4xy^{\prime \prime}+2y^{\prime}+y=0

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