Euler Approximation

• Numerical approach for approximating the solution of a DEQ
• We can use local linearity to estimate a value near the point of tangency
• Euler’s method uses repeated tangent line segments to approximate the actual solution curve

Usage

• Given $\frac{dy}{dx}=f(x,y),y(x_0)=y_0$
• $y_{i+1}=y_i+hf(x_i,y_i)$
• Where $h$ is step size (how much $x$ increases each iteration)

Error

• The Taylor series for the equation gives:
  • $\frac{y_{i+1}-y_i}{h}=y^{\prime }(x_i)+\frac{1}{2}h{y}^{\prime \prime }(x_i)+\cdots +\frac{h^{n-1}{y}^{(n)}(x_i)}{n!}$
• And Euler’r method uses:
  • $\frac{y_{i+1}-y_i}{h}=y^{\prime }(x_i)$
• The Euler method is a first-order method because it truncates all terms of $O(h)$