series solution xy^{\prime \prime}+y^{\prime}-y=0

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`x`²	`x`^□	log_□	√☐	^□√☐	≤	≥	□□	·	÷	`x`^◦	π
(☐)^′	`ddx`	∂∂`x`	∫	∫_□^□	lim	∑	∞	`θ`	(`f` ◦ `g`)	`f`(`x`)

☐²

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ddx

≥	≤	·	÷	`x`^◦	(☐)	\|☐\|	(`f` ◦ `g`)	`f`(`x`)	ln	`e`^☐
(☐)^′	∂∂`x`	∫_□^□	lim	∑	sin	cos	tan	cot	csc	sec

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∫ x²3^−xdx

Solution

−3^−xx²ln(3) −2ln²(3) · 3^−xx−2ln³(3) · 3^−x+C

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∫ x²· 3^−xdx

Apply u-substitution: ∫ −3^uu²du

=∫ −3^uu²du

Take the constant out: ∫a·f(x)dx=a·∫f(x)dx

=−∫ 3^uu²du

Apply Integration By Parts: 3^uu²ln(3) −∫ 2· 3^uuln(3) du

=−(3^uu²ln(3) −∫ 2· 3^uuln(3) du)

∫ 2· 3^uuln(3) du=2ln(3) (3^uuln(3) −3^uln²(3) )

=−(3^uu²ln(3) −2ln(3) (3^uuln(3) −3^uln²(3) ))

Substitute back u=−x

=−(3^−x(−x)²ln(3) −2ln(3) (3^−x(−x)ln(3) −3^−xln²(3) ))

Simplify −(3^−x(−x)²ln(3) −2ln(3) (3^−x(−x)ln(3) −3^−xln²(3) )): −3^−xx²ln(3) −2ln²(3) · 3^−xx−2ln³(3) · 3^−x

=−3^−xx²ln(3) −2ln²(3) · 3^−xx−2ln³(3) · 3^−x

Add a constant to the solution

=−3^−xx²ln(3) −2ln²(3) · 3^−xx−2ln³(3) · 3^−x+C

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

series solution xy^{\prime \prime}+y^{\prime}-y=0

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series solution xy^{\prime \prime}+y^{\prime}-y=0

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