Step By Step Calculus » 16.4 - Direction Fields, Euler's Method

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Synopsis
Finding an exact solution to a differential equation is not always possible. Instead, there are geometrical methods (such as direction field) and numerical methods (such as Euler’s method) to obtain approximate solutions to the differential equations.
Direction field is a graph obtained by drawing small line segments of slope f(x,y)f(x,y) at points (x,y)(x,y) for the differential equation y'=f(x,y).y'=f(x,y). It’s the graph for where one can easily visualize the general shape of the solution curves. One can use the following procedure for the direction fields.
Step 0 (Convert) Change expression to the form y'=F(x,y)y'=F(x,y).
Step 1 (Mesh) Pick “Mesh” of x-yx-y points to evaluate F(x,y)F(x,y) at.
Step 2 (Tabulate) Find and tabulate F(x,y)F(x,y) at all mesh points.
Step 3 (Direction Field) Draw line segments centered at each mesh point (x,y)(x,y) with slope F(x,y)F(x,y). We draw an arrow with each line segment to illustrate the curve’s increasing or decreasing or constant behaviour (if slope is positive then we use upward arrow to show that it is increasing, if slope is negative then we use downward arrow to show that it is decreasing, if slope is zero then we draw a horizontal line with right arrow to show that it is constant).
Step 4 (Connect) Connect the line segments in a smooth manner starting from a given point and going in both directions.
The constant solutions of the form y(x)=cy(x)=c for some constant cc to a differential equation y'=f(x,y)y'=f(x,y) are called equilibrium solutions. If all the solutions close the equilibrium solution approaches towards the constant solution as x\to \inftyx\to \infty, the equilibrium solution is stable; however, if all the solutions close to the equilibrium solution diverges from it, then the equilibrium solution is unstable.
Euler’s method approximates the solution curve by estimating the points on the solution curve using the tangent line approximation with

\displaystyle { \textbf{(Euler's Method Iterating Formula)}\qquad y(x_{i+1})\approx y_{i+1}= y_i+hf(x_i,y_i), }\displaystyle { \textbf{(Euler's Method Iterating Formula)}\qquad y(x_{i+1})\approx y_{i+1}= y_i+hf(x_i,y_i), }
\qquad\qquadfor i=0,1,2,\cdotsi=0,1,2,\cdots.

where hh is the step size and x_0, y_0x_0, y_0 are the given initial conditions.