Differential equations overview (HL)

This continues the discussion from solving simple differential equations (SL).

What are they?

A differential equation relates functions and their derivatives (and higher derivatives). Their solutions are functions instead of values.

A common application for differential equation is in physics, relating position as influenced by force, which is mass times acceleration. The differential equation will be provided to you on exams.

Previously we looked at

which can be solved by repeated integration, as long as we are provided with $n$ initial or boundary conditions.

In single-variable calculus, we will only look at ordinary differential equations (ODEs).

Notation

Depending on context, $y^\prime$ could be $y^\prime(t)$ or $y^\prime(x)$. Usually if there is no $t$, then $\displaystyle y^\prime = y^\prime(x) = \frac{\d y}{\d x}$.

$x^\prime$ is usally $x^\prime(t)$

$x$ vs. $t$

The derivatives are almost always with respect to $x$ to $t$, but it’s their locations in the derivatives that matter, not the letters themselves. Note that

can be easily solved via repeated integration, whereas

are generally much harder to solve.

Techniques

Numeric solutions in the form of Euler’s method, and Maclaurin series solutions will have question indicate as such. The other methods required in AA HL are below