What is the solution for (d^4y)/(dx^4)-a(d^2y)/(dx^2)-by=0 ?

The solution for (d^4y)/(dx^4)-a(d^2y)/(dx^2)-by=0 is

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(d^4y)/(dx^4)-a(d^2y)/(dx^2)-by=0

\frac{d ^{4} y}{d x ^{4}} - a \frac{d ^{2} y}{d x ^{2}} - b y = 0

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

Solution steps

\frac{d ^{4} y}{d x ^{4}} - a \frac{d ^{2} y}{d x ^{2}} - b y (x) = 0

Solve linear ODE:

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

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

x^{^{''}} + 2 x^{^{'}} + 0.75 x = 0

y^{^{''}} = 7 y^{^{'}} - 12 y, y (0) = 0, y^{^{'}} (0) = - 1

u^{^{''}} + 5 H (t) = 0

y^{^{''}} - 10 y^{^{'}} + 2 y = 0, y (0) = 0

\frac{d ^{4} y}{d x ^{4}} - 2 y^{^{''}} + 1 y = 0

What is the solution for (d^4y)/(dx^4)-a(d^2y)/(dx^2)-by=0 ?
The solution for (d^4y)/(dx^4)-a(d^2y)/(dx^2)-by=0 is