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 or relations instead of individual values. In single-variable calculus, we will only look at ordinary differential equations (ODEs).

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 Math AA exams.

Previously we looked at

\frac{\d ^n}{\d x^n} f(x) = g(x)

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

Order and initial values

The order of a differential equation refers to the highest derivative that appears. Most differential equations in Math AA are first order, as it involves only the first derivative. Most higher-order differential equations in Math AA are solved via repeated integration as aforementioned.

An infinite number of functions or relations solve each differential equation. Instead, $n$ initial values are needed to pinpoint the specific function or relation to solve an $n$ th-order differential equation.

Higher order differential equation may also involve boundary conditions, which can be given for different $x$ values, whereas initial values give $y$ and all values up to the $n-1$ th derivative for the same value of $x$ .

The polynomial degree for certain differential equations refer to the exponent of the highest derivative. Eg

\left(\frac{\d y}{\d x}\right)^3 + 4y^5 = \sin x

has an order of $1$ (for first derivative) and polynomial degree of $3$ (for highest derivative to the third power). So order and degree are distinct concepts in differential equations. Here one initial value is needed to pinpoint the specific solution, because of the order of $1$ .

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 $x^\prime(t)$

$\bm x$ vs. $\bm t$

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

\frac{\d ^nx}{\d t^n} = f(t)

can be easily solved via repeated integration, whereas

\frac{\d ^nx}{\d t^n} = f(x)

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

form	technique
$\displaystyle \frac{\d ^ny}{\d x^n} = f(x)$	repeated integration with respect to $x$
$\displaystyle \frac{\d y}{\d x} = \frac{f(x)}{g(y)}$	separable ODE
$\displaystyle \frac{\d y}{\d x} = Q(x) - P(x)y$	integrating factor
$\displaystyle \frac{\d y}{\d x} = f\left(\frac yx\right)$	let $y = vx$

No matter the analytical technique in Math AA, you are one way or another need to integrate to cancel the derivative to solve.

Integral of $\frac{1}{x}$

One of the first antiderivatives you learned is

\int \frac1x \d x = \ln \vert x\vert + C

However if you ever need raise $\e$ to this exponent, then

y = \e^{\ln \vert x\vert + C}

is two curves, meaning both $y = \e^C x$ and $y = -\e^C x$ . Which curve is right depends on the initial value given. In most cases, we can just group it as $y = Ax$ , and this allows for both positive and negative $A$ .