What is the solution for y^{''}-6y^'-27y=0,y(0)=α,y^'(0)=21 ?

The solution for y^{''}-6y^'-27y=0,y(0)=α,y^'(0)=21 is y=e^{9t}(α+(7-3α)/4)-(e^{-3t}(7-3α))/4

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y^{''}-6y^'-27y=0,y(0)=α,y^'(0)=21

y^{^{''}} - 6 y^{^{'}} - 27 y = 0, y (0) = α, y^{^{'}} (0) = 21

y = e^{9 t} (α + \frac{7 - 3 α}{4}) - \frac{e ^{- 3 t} ( 7 - 3 α )}{4}

Solution steps

y^{''} (t) - 6 y^{'} (t) - 27 y = 0

Solve linear ODE:

y = e^{9 t} (α + \frac{7 - 3 α}{4}) - \frac{e ^{- 3 t} ( 7 - 3 α )}{4}

y = e^{9 t} (α + \frac{7 - 3 α}{4}) - \frac{e ^{- 3 t} ( 7 - 3 α )}{4}

f^{^{''}} (t) - 4 f^{^{'}} (t) + 3 f (t) = 0, f (0) = 3, f^{^{'}} (0) = 5

\frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} + 7 y = 0

5 y^{^{''}} + 50 y^{^{'}} + 125 y = 0

\frac{d ^{3} y}{d x ^{3}} - \frac{d y}{d x} = 0

y^{^{''}} + a y = 0, y (0) = a, y^{^{'}} (0) = 0

What is the solution for y^{''}-6y^'-27y=0,y(0)=α,y^'(0)=21 ?
The solution for y^{''}-6y^'-27y=0,y(0)=α,y^'(0)=21 is y=e^{9t}(α+(7-3α)/4)-(e^{-3t}(7-3α))/4