Question - Finding Differential Equation for General Solution
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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