The given equation is a linear homogeneous differential equation of order 4. Let's analyze and solve it step by step.

Step 1: General form of the equation

The equation is: $2 \frac{d^4y}{dx^4} - 8 \frac{d^3y}{dx^3} - 11 \frac{d^2y}{dx^2} + 66 \frac{dy}{dx} - 45y = 0$

We assume a solution of the form: $y = e^{rx}$ where $r$ is a constant to be determined. Substituting $y = e^{rx}$ into the equation transforms it into the characteristic equation.

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Step 2: Characteristic equation

Substituting $y = e^{rx}$ and its derivatives into the equation, we get: $2r^4 - 8r^3 - 11r^2 + 66r - 45 = 0$

This is a quartic equation in $r$. We solve it by factoring or using numerical methods.

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Step 3: Factoring the characteristic equation

Let's factorize $2r^4 - 8r^3 - 11r^2 + 66r - 45 = 0$. By trial or synthetic division, we find that $r = 3$ is a root.

Using $(r - 3)$ as a factor, perform synthetic or long division on the polynomial:

Division of $2r^4 - 8r^3 - 11r^2 + 66r - 45$ by $(r - 3)$:

$2r^4 - 8r^3 - 11r^2 + 66r - 45 \div (r - 3) = 2r^3 - 2r^2 - 17r + 15$

Now, the equation becomes: $(r - 3)(2r^3 - 2r^2 - 17r + 15) = 0$

Further factoring $2r^3 - 2r^2 - 17r + 15$:

Using trial roots again, $r = -1$ is a root.

Divide $2r^3 - 2r^2 - 17r + 15$ by $(r + 1)$: $2r^3 - 2r^2 - 17r + 15 \div (r + 1) = 2r^2 - 4r - 15$

Now, the equation becomes: $(r - 3)(r + 1)(2r^2 - 4r - 15) = 0$

Factorizing $2r^2 - 4r - 15$:

Solve $2r^2 - 4r - 15 = 0$ using the quadratic formula: $r = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-15)}}{2(2)}$ $r = \frac{4 \pm \sqrt{16 + 120}}{4} = \frac{4 \pm \sqrt{136}}{4} = \frac{4 \pm 2\sqrt{34}}{4} = 1 \pm \frac{\sqrt{34}}{2}$

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Step 4: General solution

The roots of the characteristic equation are: $r = 3, \quad r = -1, \quad r = 1 + \frac{\sqrt{34}}{2}, \quad r = 1 - \frac{\sqrt{34}}{2}$

Thus, the general solution is: $y(x) = C_1 e^{3x} + C_2 e^{-x} + C_3 e^{\left(1 + \frac{\sqrt{34}}{2}\right)x} + C_4 e^{\left(1 - \frac{\sqrt{34}}{2}\right)x}$ where $C_1, C_2, C_3,$ and $C_4$ are arbitrary constants.

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Do you need further clarification or details? 😊

5 Related Questions:

1. How do we verify that the solution satisfies the original differential equation?
2. What happens to the solution if one root has multiplicity greater than 1?
3. Can this equation be solved numerically without factoring?
4. How does the solution change if the equation becomes inhomogeneous?
5. How are complex roots handled in differential equations?

Tip:

When solving higher-order differential equations, always check if the characteristic polynomial can be factored to simplify the process.