What is expand y''(x)(3+1/x)^3 ?

The solution to expand y''(x)(3+1/x)^3 is 27y''(x)+(27y''(x))/x+(9y''(x))/(x^2)+(y''(x))/(x^3)

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expand y''(x)(3+1/x)^3

expand

y^{''} (x) (3 + \frac{1}{x})^{3}

27 y'' (x) + \frac{27 y '' ( x )}{x} + \frac{9 y '' ( x )}{x ^{2}} + \frac{y '' ( x )}{x ^{3}}

Solution steps

y^{''} (x) (3 + \frac{1}{x})^{3}

(3 + \frac{1}{x})^{3} = 27 + \frac{27}{x} + \frac{9}{x ^{2}} + \frac{1}{x ^{3}}

= y^{^{''}} (x) (\frac{27}{x} + \frac{9}{x ^{2}} + \frac{1}{x ^{3}} + 27)

Distribute parentheses

= y^{^{''}} (x) 27 + y^{^{''}} (x) \frac{27}{x} + y^{^{''}} (x) \frac{9}{x ^{2}} + y^{^{''}} (x) \frac{1}{x ^{3}}

= 27 y^{^{''}} (x) + \frac{27}{x} y^{^{''}} (x) + \frac{9}{x ^{2}} y^{^{''}} (x) + \frac{1}{x ^{3}} y^{^{''}} (x)

Simplify

27 y^{''} (x) + \frac{27}{x} y^{''} (x) + \frac{9}{x ^{2}} y^{''} (x) + \frac{1}{x ^{3}} y^{''} (x) : 27 y^{''} (x) + \frac{27 y ^{''} ( x )}{x} + \frac{9 y ^{''} ( x )}{x ^{2}} + \frac{y ^{''} ( x )}{x ^{3}}

= 27 y^{''} (x) + \frac{27 y ^{''} ( x )}{x} + \frac{9 y ^{''} ( x )}{x ^{2}} + \frac{y ^{''} ( x )}{x ^{3}}

(e^{i wt} - e^{- i wt}) e^{- in wt}

(72 - 3) (53 + 2)

\frac{- x ^{2} - 2 x + 1}{( x ^{2} + 1 ) ^{2}}

(e^{t} - 9)^{2}

(- 9 sin (x) + 9 sin (9 x))^{2}

What is expand y''(x)(3+1/x)^3 ?
The solution to expand y''(x)(3+1/x)^3 is 27y''(x)+(27y''(x))/x+(9y''(x))/(x^2)+(y''(x))/(x^3)