Question

Graph the isoclines of the ODE y' = y + t² where the slope is – 1,0,1.

Answer 1

To graph the isoclines of the ordinary differential equation (ODE)
`y^(') = y + t^(2)`

where the slope is -1, 0, and 1, we'll set the right-hand side of the equation equal to each of these slopes and solve for y. This will give us curves where the slope of the solution is constant.

Let's solve for the isoclines:

For slope -1:

`y^(') = -1`

`y + t^(2) = -1`

`y = -1 - t^(2)`

For slope 0:

`y^(') = 0`

`y + t^(2) = 0`

`y = - t^(2)`

For slope :

`y^(') = 1`

`y + t^(2) = 1`

`y = 1 - t^(2)`

Now, you can graph these curves on a coordinate system and you'll have the isoclines for the given ODE with slopes of -1, 0, and 1.