You are asked to generate a slope field for the initialvalue

You are asked to generate a slope field for the initial-value problem The slope field allows us to obtain an approximate solution graphically. Answer the following 5 questions. What is the slope of the tangent line that passes through the initial value? What can we say about the function y at the initial value? Circle one. Which is more appropriate description of the solution that passes through the initial value? Circle one Let us consider a range of values on the x-y plane: x=1, but y can take any values between -2 and 2. Which of the statements are true ? Slope would be constant for all values of y described above Slope can be computed, but value is different depending on y values Slope cannot be computed for some of the y values noted above True or false: in order to compute the slope, we need to express, as an explicit function of x, i.e., we need to obtain y=y(x).

Solution

(a) Expression for slope would be 1-x+y². But since we need to find slope of the tangent line that passes through initial value. Therefore, we need to plug x=2 and y=0 to get the required slope.

Slope = 1 - 2 + 0²

Slope = -1

(b) Since slope is negative at initial value, therefore, we can say that function is decreasing at that point. :-)

(c) Particular solution is always a better description of a solution of any differential equation at the given initial point. Basically, if we do not use the given initial value, then we get the general solution. General solution do not describe anything about initial value. Use find the values of constants in the general solution using the given initial value and hence we get the particular solution, therefore, particular solution best decribes the characteristic of an initial value problem as compared to general solution.

(d) We know that slope of the function is given to us as 1-x+y². When x=1, the expression for slope becomes:

Slope = 1 - 1 + y² = y²

Therefore, we can see that slope is a function of y, but it is a quadratic function of y. Thefore on the interval (-2, 2) slope will have positive sign. Since slope depends on y, therefore, it will be different depending on the values of y. Hence option B is most appropriate.

(e) This statement is false. In order to complute slope, we alway need to expression y\' (i.e. dy/dx) in terms of x and y.