4. Consider the differential equation = 2x - y. (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point (0, 1). (Note: Use the axes provided in the pink test booklet.) (b) The solution curve that passes through the point (0, 1) has a local minimum at x In What is the y-coordinate of this local minimum? (c) Let y= f(x) be the particular solution to the given differential equation with the initial condition (0) = 1, Use Euler's method, starting at x 0 with two steps of equal size, to approximate f(-0.4). Show the work that leads to your answer. (d) Find de d'y in terms of x and y, Determine whether the approximation found in part (c) is less than or greater than f(-0.4). Explain your reasoning.

Advanced Engineering Mathematics
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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
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Please help with 4 (particularly b-d)

**Question 4: Differential Equation Analysis**

Consider the differential equation \(\frac{dy}{dx} = 2x - y\).

(a) **Slope Field and Solution Curve Sketching**

   On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point (0, 1).

   (Note: Use the axes provided in the pink test booklet.)

   *Diagram Explanation:* The diagram shows a set of short line segments (slopes) plotted at specific grid points. These slopes represent the direction of the solution curve at each point. The arrows indicate the general direction of the curve.

(b) **Local Minimum Analysis**

   The solution curve that passes through the point (0, 1) has a local minimum at \(x = \ln\left(\frac{3}{2}\right)\). What is the \(y\)-coordinate of this local minimum?

(c) **Euler’s Method for Approximation**

   Let \(y = f(x)\) be the particular solution to the given differential equation with the initial condition \(f(0) = 1\). Use Euler’s method, starting at \(x = 0\) with two steps of equal size, to approximate \(f(-0.4)\). Show the work that leads to your answer.

(d) **Second Derivative Analysis**

   Find \(\frac{d^2y}{dx^2}\) in terms of \(x\) and \(y\). Determine whether the approximation found in part (c) is less than or greater than \(f(-0.4)\). Explain your reasoning.
Transcribed Image Text:**Question 4: Differential Equation Analysis** Consider the differential equation \(\frac{dy}{dx} = 2x - y\). (a) **Slope Field and Solution Curve Sketching** On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point (0, 1). (Note: Use the axes provided in the pink test booklet.) *Diagram Explanation:* The diagram shows a set of short line segments (slopes) plotted at specific grid points. These slopes represent the direction of the solution curve at each point. The arrows indicate the general direction of the curve. (b) **Local Minimum Analysis** The solution curve that passes through the point (0, 1) has a local minimum at \(x = \ln\left(\frac{3}{2}\right)\). What is the \(y\)-coordinate of this local minimum? (c) **Euler’s Method for Approximation** Let \(y = f(x)\) be the particular solution to the given differential equation with the initial condition \(f(0) = 1\). Use Euler’s method, starting at \(x = 0\) with two steps of equal size, to approximate \(f(-0.4)\). Show the work that leads to your answer. (d) **Second Derivative Analysis** Find \(\frac{d^2y}{dx^2}\) in terms of \(x\) and \(y\). Determine whether the approximation found in part (c) is less than or greater than \(f(-0.4)\). Explain your reasoning.
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