Solve Equation

Given,

Y'''' - 2 Y ''' + 2 Y'' - 2Y' + Y = 0 ---(1)

Theorems to be used.
a) y' ---> d/dx (y)
b) d/dx (Y^n) = n Y ^ (n-1)
c) Integration of (Y') = Y
d) Integration of (Y^n) = (Y^n+1)/n+1
e) Integration of (Constant) = Y

Integrating (1) by using the above theorems

Y''' - 2 Y '' + 2 Y' - 2Y + (y^2)/2 = 0 ---(2)

Integrating (2) by using the above theorems

Y'' - 2 Y ' + 2 Y - Y^2 + (y^3)/6 = 0 ---(3)
Integrating (3) by using the above theorems

Y' - 2 Y + Y^2 - Y^3/6 + (Y^4)/24 = 0 ---(4)

Integrating (4) by using the above theorems

Y - Y^2 + Y^3/3 - Y^4/24 + (y^5)/120 = 0 ---(5)

Solving the above equation,

Y^5 - 5Y^4 + 40 Y^3 - 120Y^2 + 120 Y = 0

This is the final answer.

Hope this helped

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