What is the general solution to the differential equation that generated the slope field? Choose 1 answer: (a) y=x+C (b) y=x^(2)+C (c) x=y^(2)+C (d) y^(2)=x^(2)+C (e) x^(2)+y^(2)=C

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Identify Differential Equation: To solve the problem, we need to identify the differential equation that corresponds to the given slope field options. A slope field represents the slopes of tangent lines to solutions of a differential equation at various points. The general solution to the differential equation will be the function whose derivative matches the slope at any given point (x, y) on the graph.Analyze Option (a): Let's analyze option (a) y = x + C. The derivative of y with respect to x is dy/dx = 1. This means that the slope at every point on the graph would be 1, which corresponds to a slope field of horizontal lines with a slope of 1. This does not give us any information about the relationship between x and y in the slope field, so we cannot confirm if this is the correct differential equation without more information.Consider Option (b): Now let's consider option (b) y = x^2 + C. The derivative of y with respect to x is dy/dx = 2x. This means that the slope at any point (x, y) on the graph would be 2x, which implies that the slope depends on the x-coordinate. This could potentially match a slope field where the slope of the lines increases as we move away from the y-axis in either direction.Examine Option (c): Next, we look at option (c) x = y^2 + C. Taking the derivative with respect to y, we get dx/dy = 2y. This means that the slope at any point (x, y) on the graph would be 2y, which implies that the slope depends on the y-coordinate. This could potentially match a slope field where the slope of the lines increases as we move away from the x-axis in either direction.Investigate Option (d): For option (d) y^2 = x^2 + C, we differentiate both sides with respect to x to get 2y(dy/dx) = 2x. This simplifies to dy/dx = x/y, assuming y is not zero. This means that the slope at any point (x, y) on the graph would be x/y, which implies that the slope depends on both the x-coordinate and the y-coordinate. This could potentially match a slope field where the slope of the lines changes depending on the position relative to both axes.Review Option (e): Finally, we consider option (e) x^2 + y^2 = C. Differentiating both sides with respect to x, we get 2x + 2y(dy/dx) = 0. This simplifies to dy/dx = -x/y, assuming y is not zero. This means that the slope at any point (x, y) on the graph would be -x/y, which implies that the slope depends on both the x-coordinate and the y-coordinate and that the slope field would consist of circles centered at the origin.Final Analysis: Without the actual slope field to compare against, we cannot definitively determine which option is correct. However, we can analyze the structure of the differential equations and their solutions. Options (a) and (b) suggest that the slope is independent of y, while options (c), (d), and (e) suggest that the slope depends on both x and y. Without additional information, we cannot proceed further in determining the correct answer.

What is the general solution to the differential equation that generated the slope field? $\newline$ Choose $1$ answer: $\newline$ $(a)$ $y=x+C$ $\newline$ $(b)$ $y=x^{2}+C$ $\newline$ $(c)$ $x=y^{2}+C$ $\newline$ $(d)$ $y^{2}=x^{2}+C$ $\newline$ $(e)$ $x^{2}+y^{2}=C$

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