From: Andrew Lorimer Date: Wed, 22 May 2019 05:00:05 +0000 (+1000) Subject: [spec] add notes for rest of chapter 9 X-Git-Tag: yr12~126 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/9db9641366c2dd17645b83cd4be32ccf62aff880 [spec] add notes for rest of chapter 9 --- diff --git a/spec/calculus.md b/spec/calculus.md index 97bc926..3ed01c3 100644 --- a/spec/calculus.md +++ b/spec/calculus.md @@ -408,3 +408,18 @@ $$\left({dm \over dt}\right)_\Sigma = \left({dm \over dt}\right)_{\text{in}} - \ If ${dy \over dx}=f(x)g(y)$, then: $$\int f(x) \> dx = \int {1 \over g(y)} \> dy$$ + +### Using definite integrals to solve DEs + +Used for situations where solutions to ${dy \over dx} = f(x)$ is not required. + +In some cases, it may not be possible to obtain an exact solution. + +Approximate solutions can be found by numerically evaluating a definite integral. + +### Using Euler's method to solve a differential equation + +$${{f(x+h) - f(x)} \over h } \approx f^\prime (x) \quad \text{for small } h$$ + +$$\implies f(x+h) \approx f(x) + hf^\prime(x)$$ +