7.5 Approximating Solutions Using Euler’s Method

Learning objectives

—FUN-7.C—Estimate solutions to differential equations—Topic 7.5• FUN-7.C Estimate solutions to differential equations.• FUN-7.C.4 Euler’s method provides a procedure for approximating a solution to a differential equation or a point on a solution curve.• Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

For $dy/dx=f(x,y)$ , Euler’s method starts from a known point $(x_n,y_n)$ and uses the tangent slope there to estimate the next point. Over a horizontal step $h$ , the estimated vertical change is slope $\times$ step, or $f(x_n,y_n)h$ .

x_{n+1}=x_n+h,\qquad y_{n+1}=y_n+h,f(x_n,y_n)

At each step: (1) evaluate the slope at the current point; (2) multiply it by the signed step size $h$ ; (3) add that change to the current $y$ ; (4) advance $x$ by $h$ ; then recompute the slope at the new point.

For $dy/dx=x+y$ , $y(0)=1$ , and $h=0.1$ : from $(0,1)$ the slope is $1$ , so $(x_1,y_1)=(0.1,1+0.1(1))=(0.1,1.1)$ . Now the slope is $0.1+1.1=1.2$ , so $(x_2,y_2)=(0.2,1.1+0.1(1.2))=(0.2,1.22)$ . Thus $y(0.2)\approx1.22$ .

Euler’s method produces an approximation, not an exact solution. Do not reuse the first slope for every step: the derivative must be evaluated at each new approximate point. A smaller step generally follows changing slopes more closely, but it also requires more steps; moving toward smaller $x$ requires a negative $h$ .

Next Topic

—7.6 Finding General Solutions Using Separation of VariablesRead