AP Calculus BC / Unit 7: Differential Equations
7.5 Approximating Solutions Using Euler's Method
Notes
Step along tangent lines: $$y_{new}=y_{old}+\frac{dy}{dx}\Big|_{(x,y)}\cdot\Delta x$$
- Repeat, updating $x$ and $y$ each step.
Practice
$\displaystyle \frac{dy}{dx}=x+y$, $y(0)=1$. One Euler step with $\Delta x=0.1$ gives $y(0.1)\approx$
Slope at $(0,1)$ is 1: $1+1(0.1)$.
A second step from $(0.1,1.1)$ gives $y(0.2)\approx$
Slope $1.2$: $1.1+0.12$.
If solutions are concave up, Euler's method gives:
Tangent lines lie below a concave-up curve.
Ready for more? Take the Unit 7 test.