What is the solution for (4D^4-45D^2-70D-24)y=0 ?

The solution for (4D^4-45D^2-70D-24)y=0 is y=c_{1}e^{-2x}+c_{2}e^{4x}+c_{3}e^{-x/2}+c_{4}e^{-(3x)/2}

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(4D^4-45D^2-70D-24)y=0

(4 D^{4} - 45 D^{2} - 70 D - 24) y = 0

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

Solution steps

(4 D^{4} - 45 D^{2} - 70 D - 24) y = 0

Solve linear ODE:

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

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

y^{^{''}} - 6 y^{^{'}} + 113 y = 0

10 \frac{d ^{2} y}{d x ^{2}} + 20 \frac{d y}{d x} + 10 y = 0

y^{^{''}} + 4 y^{^{'}} + 3 y = 0, y (0) = 6, y^{^{'}} (0) = 2

x^{^{''}} + 5 x^{^{'}} = 0

\frac{d ^{2} y}{d x ^{2}} - cy = 0

What is the solution for (4D^4-45D^2-70D-24)y=0 ?
The solution for (4D^4-45D^2-70D-24)y=0 is y=c_{1}e^{-2x}+c_{2}e^{4x}+c_{3}e^{-x/2}+c_{4}e^{-(3x)/2}