00:01
We are given a differential equation.
00:03
We are asked to use this equation to graph the slope field and three solutions for the equation.
00:13
The equation we're given is that y prime is equal to one half y.
00:20
We're going to draft the slope field.
00:23
Let's consider what the slope will be at different points x y.
00:31
So, in particular, if we take y to be, some constant c in the real numbers and x is any real number then you have that slope of y at the point x c is going to be one half y which is equal to one half of c so in other words say if c is zero you have that the slope of y is going to be 0 at every point along the x -axis.
02:05
So in particular, plugging the different values of c, we have that the slope for any y equals c is going to be 0.
02:17
So along the x we have horizontal lines indicating a slope of 0.
02:29
If y is equal to say 0 .5, we'll have a slope of 0 .5.
02:37
Y is equal to 1, we'll have a slope of 1.
02:40
Why is it a 1 .5 will have a slope of 0 .a.
02:46
Y is equal to 0 .5, we'll have a slope of 0 .25.
02:55
So it's just barely above horizontal.
02:58
Y is equal to 1 will have a slope of 0 .5.
03:01
Y is equal to 1 .5, we'll have a slope of 3 quarters.
03:06
Why is equal to 2, we have a slope of 1.
03:10
And you see that the slope simply increases as you go up the y -axis.
03:19
You can extend each of these from the one axis for all values of x.
03:37
It's a little bit of a rough drawing.
03:41
The idea is that as you go further away from the origin, like these slopes get steeper and steeper...