What is the solution for (D^5-D^3)y=0 ?

The solution for (D^5-D^3)y=0 is y=c_{1}e^{-x}+c_{2}e^x+c_{3}+c_{4}x+c_{5}x^2

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(D^5-D^3)y=0

(D^{5} - D^{3}) y = 0

y = c_{1} e^{- x} + c_{2} e^{x} + c_{3} + c_{4} x + c_{5} x^{2}

Solution steps

(D^{5} - D^{3}) y = 0

Solve linear ODE:

y = c_{1} e^{- x} + c_{2} e^{x} + c_{3} + c_{4} x + c_{5} x^{2}

y = c_{1} e^{- x} + c_{2} e^{x} + c_{3} + c_{4} x + c_{5} x^{2}

y^{^{''}} - 4 y^{^{'}} + 4 y = 0, y (0) = 0, y^{^{'}} (0) = 4

y^{^{''}} - 9 y^{^{'}} + 20 y = 0, y (0) = 3, y^{^{'}} (0) = 2

y^{^{''}} + \frac{1}{100} y^{^{'}} = 0

x^{^{''}} - 7 x^{^{'}} + 12 = 12, x (0) = 5, x^{^{'}} (0) = 5

y^{^{''}} - 4 y = 0, y (0) = 2, y^{^{'}} (0) = 4

What is the solution for (D^5-D^3)y=0 ?
The solution for (D^5-D^3)y=0 is y=c_{1}e^{-x}+c_{2}e^x+c_{3}+c_{4}x+c_{5}x^2