Final answer:
To derive the differential equation relating y(t) to X(t) from the given transfer function, perform an inverse Laplace transform and equate it to the output Y(s), which results in a polynomial in s representing the output transformed.
Step-by-step explanation:
The correct answer is to find the differential equation relating the output y(t) to the input X(t) based on the given transfer function H(s). First, the transfer function is given in the Laplace domain by H(s) = ​(2s^2 + 5s + 7). To find the differential equation, we can take the inverse Laplace transform of H(s).
We equate H(s) to Y(s)/X(s), where Y(s) and X(s) are the Laplace transforms of y(t) and X(t), respectively. Multiplying both sides by X(s) and then taking the inverse Laplace transform gives us a differential equation in the time domain.
To solve for the differential equation, we express H(s)X(s) as a polynomial in s representing the transformed output, as Y(s) = H(s)X(s).
This polynomial can be equated to the coefficients in H(s) times derivatives of y(t) if we know the form of X(t), or it can be left in terms of the derivatives of x(t). The resulting differential equation is usually a second-order or higher-order equation, depending on the degree of H(s).
The correct answer is option (e) y = -2/(x^3).
To sketch the graph of y = -2/(x^3), you can start by identifying the key features of the equation. The graph will be symmetric about both the x-axis and the y-axis, and it will pass through the point (1, -2). As x approaches 0 from the left or right, y approaches negative or positive infinity respectively. As x approaches positive or negative infinity, y approaches 0.
Using this information, you can plot the graph. Start by plotting the point (1, -2). Then, draw the curve passing through this point and approaching the x-axis as x approaches infinity in both the positive and negative directions.