Consider the differential equation

Question

A graphing calculator may NOT be used for the following problem

Consider the differential equation dy/dx = x(y - 1).

a) On the axes provided, sketch a slope field for the given differential equation at the 8 given points.

The 8 points are: (0, 2) (1, 2) (0, 1) (1, 1) (0, 0) (1, 0) (0, -1) (-1, -1)

b) Let y = f(x) be the particular solution to the given differential equation with the initial condition f(1) = 2. Write an equation for the line tangent to the graph of y = f(x) at x = 1. Use your equation to approximate f(1.5).

c) Find the particular solution y = f(x) to the given differential equation with the initial condition f(1) = 2.

Answer 1

The previous answer provided was ln(y-1)=((x²)/2)-1/2

to make more clear y is function of x

can rewrite y(x)=e^x*x/2-1/2+1,,,,,where x*x/2=(x²)/2

y(x)=e^-1/2*e^x*x/2+1

update: it is neat that dy/dx = e^-1/2(xe^x*x/2)=x(y-1)

at x=1, dy/dx=1 from original given

point is 1,2

1=(y-2)/(x-1)

y=x+1

at x=1.5 y=2.5

for slope field, any time x=0 or y=1, slope is 0 ,,,draw little horiuzontal line.

,,,the other points draw in little slope line.