[spec] add notes for rest of chapter 9

diff --git a/spec/calculus.md b/spec/calculus.md
index 97bc926a7cb6b59e66ec2eba6c634ac742cb6028..3ed01c3fa748c39df9f0f503d883f99f11d24975 100644 (file)
--- 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:
 
 $$\int f(x) \> dx = \int {1 \over g(y)} \> dy$$
 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
+
+$${{f(x+h) - f(x)} \over h } \approx f^\prime (x) \quad \text{for small } h$$
+
+$$\implies f(x+h) \approx f(x) + hf^\prime(x)$$
+