Question

sketch the gradient field for the differential equation dy/dt=t-y² for t=-1......,1

y=-1,.......,1

Answer 1

The gradient field (Slope field or Directions field) is a graphical representation of the solutions to a first-order differential equation (ODE) of a scalar function. The gradient field shows the slope of a differential equation at specific vertical and horizontal intervals on a plane and can be used to determine the approximate slope of a tangent at a point on a curve, where the curve is some solution to the differential equation.

To construct a gradient field, it is necessary to calculate the slope of the tangent to the curve of the solution of the ODE at each point of the plane.

Solution for y'=t-y²:

Since the left side of the ODE corresponds to the slope of the tangent, substituting the coordinates of a point on the plane into the right side of the ODE, we obtain the slope of the tangent.

For example, for point (t₀,y₀)=(0.5, 0.5):

$$m=\frac {dy}{dt}=t-y^2=0.5-0.5^2=0.25$$

Let's set a small increment for t:

$$t_1 =t_0+\delta t=0.5+0.01=0.51$$

then from the equation of a straight line on the plane we obtain the value y₁:

$$y_1 = y_0 +m(t_1-t_0)=0.5+0.25(0.01)=0.5025$$

it remains to build a vector connecting the points (0.5,0.5) and (0.51,0.5025).

Repeating for each point on the plane, we get a gradient field.

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