To solve this equation, we need to find the value of Y.

First, let's simplify the left side of the equation:

(1/2x^3 + 1/3x^4 + 1/4x^5 +...+ 1/99x^100) x Y = 1/5

To simplify the expression, we can factor out 1/x^3 from each term:

(1/x^3)(1/2 + 1/3x + 1/4x^2 +...+ 1/99x^97) x Y = 1/5

Now, let's simplify the expression inside the parentheses:

1/2 + 1/3x + 1/4x^2 +...+ 1/99x^97

We can rewrite this as a sum of fractions with a common denominator:

(1/2)(x^97/x^97) + (1/3)(x^96/x^96) + (1/4)(x^95/x^95) +...+ (1/99)(x/x)

= (x^97/2x^97) + (x^96/3x^96) + (x^95/4x^95) +...+ (x/99x)

= (1/2) + (1/3) + (1/4) +...+ (1/99x)

Now, let's substitute this back into the original equation:

(1/x^3)(1/2 + 1/3x + 1/4x^2 +...+ 1/99x^97) x Y = 1/5

(1/x^3)((1/2) + (1/3) + (1/4) +...+ (1/99x)) x Y = 1/5

Now, we can cancel out the common factors:

(1/x^3)(1/5) = (1/2) + (1/3) + (1/4) +...+ (1/99x)

To solve for Y, we can multiply both sides of the equation by x^3:

1/5 = (1/2) + (1/3) + (1/4) +...+ (1/99x) * x^3

1/5 = (1/2) + (1/3) + (1/4) +...+ (1/99) * x^2

Now, we can solve for x by finding the common denominator of the fractions on the right side:

1/5 = (1/2) + (1/3) + (1/4) +...+ (1/99) * x^2

1/5 = (99/198) + (66/198) + (49/198) +...+ (2/198) * x^2

1/5 = (99 + 66 + 49 +...+ 2)/198 * x^2

1/5 = (1 + 2 + 3 +...+ 99)/198 * x^2

1/5 = (99 * 100/2)/198 * x^2

1/5 = (99 * 50)/198 * x^2

1/5 = 4950/198 * x^2

1/5 = 25/1 * x^2

1/5 = 25x^2

Now, we can solve for x:

x^2 = (1/5) / 25

x^2 = 1/125

x = ±√(1/125)

x = ±1/√125

x = ±1/5√5

Therefore, the value of Y is ±5√5.