Example Question - constant coefficients
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Solving Second Order Homogeneous Differential Equations with Constant Coefficients
Это линейное однородное дифференциальное уравнение второго порядка с постоянными коэффициентами. Чтобы найти его решение, нам нужно сначала найти характеристическое уравнение, которое выглядит следующим образом: r^2 - 8r + 16 = 0 Теперь решим это квадратное уравнение: (r - 4)^2 = 0 Это уравнение имеет корень кратности два: r = 4. Поскольку у нас есть кратный корень, общее решение данного дифференциального уравнения будет комбинацией линейно-независимых решений вида: y(x) = C_1e^(4x) + C_2xe^(4x), где C_1 и C_2 — произвольные константы, определяемые начальными или граничными условиями задачи.
Finding Differential Equation for General Solution
The image contains a handwritten request to find the differential equation associated with the general solution `y = Be^{(αx + β)}`, where α, β, and B are constants. To find the differential equation, we need to eliminate the constants α, β, and B from the given general solution. Given: \[ y = Be^{(αx + β)} \] Step 1: Differentiate both sides with respect to x. \[ \frac{dy}{dx} = Be^{(αx + β)} \cdot α \] Step 2: We need a second derivative since there are two independent constants (α and β), so differentiate once more with respect to x. \[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( Be^{(αx + β)} \cdot α \right) = Be^{(αx + β)} \cdot α^2 \] Step 3: Now, we need to eliminate B from these equations. From the first derivative, isolate B: \[ B = \frac{1}{α} \frac{dy}{dx} \cdot e^{-(αx + β)} \] Step 4: Substituting B from the equation obtained from the first derivative into the second derivative, we get: \[ \frac{d^2y}{dx^2} = \frac{1}{α} \frac{dy}{dx} \cdot e^{-(αx + β)} \cdot α^2 \cdot e^{(αx + β)} \] Step 5: Cancel out the exponential terms \( e^{-(αx + β)} \cdot e^{(αx + β)} \) which is equal to 1. This gives us: \[ \frac{d^2y}{dx^2} = α \frac{dy}{dx} \] Step 6: We still need to eliminate α and β. Since β does not directly appear in the derived differential equation, we're left with only eliminating α. To eliminate α, we observe that our first derivative is: \[ \frac{dy}{dx} = α y \] Step 7: Divide the second derivative by y: \[ \frac{1}{y} \frac{d^2y}{dx^2} = α^2 \] Step 8: From the relation in Step 6 where we have \( \frac{dy}{dx} = α y \), we can write α as: \[ α = \frac{1}{y} \frac{dy}{dx} \] Step 9: Substituting this value of α back into the Step 7 equation: \[ \frac{1}{y} \frac{d^2y}{dx^2} = \left( \frac{1}{y} \frac{dy}{dx} \right)^2 \] Step 10: This is the desired second-order homogeneous differential equation with constant coefficients, and it is free from the constants α, β, and B: \[ \frac{d^2y}{dx^2} = \left( \frac{dy}{dx} \right)^2 / y \] Or equivalently, \[ y \frac{d^2y}{dx^2} - \left( \frac{dy}{dx} \right)^2 = 0 \] This is the differential equation associated with the given general solution.
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