Unit 8 - Advanced Calculus (HL only)
Required Prior Knowledge
Questions
Find the derivatives of the following:
a) \(xy^{2}\)
b) \(\frac{x}{y}\)
c) \(x^{2}+xy\)
d) \(\frac{y^2}{x}\)
Solutions
Get Ready
Questions
Find the Maclaurin Series expansion for$$f\left(x\right)=\frac{1}{x^{2}+1}$$by using the fact that $$\frac{d}{dx}\left(\arctan x\right)=\frac{1}{x^{2}+1}$$
Find the Maclaurin Series expansion for $$g\left(x\right)=\frac{1}{1+x}$$by finding an appropriate derivative.
Solutions
Notes
We can sometimes use the Maclaurin Series expansion for a function to approximate the general solution to a differential equation.
Suppose you have a differential equation of the form$$\frac{dy}{dx}=F\left(x,y\right)\qquad y\left(0\right)=y_{0}$$with solution \(y=f\left(x\right)\).
Then we can write \(f\left(x\right)\) as $$f\left(x\right)=f\left(0\right)+f’\left(0\right)x+\frac{f’\left(0\right)}{2!}x^{2}+\frac{f’’\left(0\right)}{3!}x^{3}+…$$
We know that \(f\left(0\right)=y_{0}\) (by the boundary condition).
We also have that $$f'\left(0\right) = \left.\frac{dy}{dx}\right|_{\substack{x=0 \\ y=y_0}} = F(0, y_0)$$
We can then find further derivatives if needed using implicit differentiation.
Examples and Your Turns
Example
Find the Maclaurin Series up to \(x^{2}\) for the solution to the differential equation$$\frac{dy}{dx}=x^{2}+y^{2}\qquad\qquad y=3\text{ when }x=0$$
Your Turn
Find the first three non-zero terms of the Maclaurin Series for the solution to the differential equation$$\frac{dy}{dx}=x-4xy\qquad\qquad y\left(0\right)=1$$
Your Turn
Find the first six terms of the Maclaurin Series for the solution to the differential equation$$\frac{dy}{dx}=y^{2}-x\qquad\qquad y\left(0\right)=1$$
Your Turn
Find the first six terms of the Maclaurin Series for the solution to the differential equation$$\frac{dy}{dx}=y\qquad\qquad y\left(0\right)=1$$
Your Turn (Challenge!)
Find the Maclaurin Series up to \(x^{3}\) for the solution to the differential equation$$\frac{d^{2}y}{dx^{2}}+y\frac{dy}{dx}=e^{x}\qquad\qquad y\left(0\right)=0, y’\left(0\right)=1$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
When using this method, which values of the Maclaurin Series are you given ‘for free’ in the question?
Is this result an exact solution to the differential equation? Is it more or less accurate than Euler’s Method?
What restrictions are there on the initial conditions to be able to use this method?
Common Mistakes / Misconceptions
The most common mistake is misapplying the Maclaurin Series formula, for example forgetting to divide by \(n!\).
Another common mistake is to do the implicit differentiation wrong to find further derivatives, because your brain is so busy with everything else going on in the question.
Connecting This to Other Skills
This is a method for approximating a polynomial solution to a Differential Equation (8.4) with initial condition given at \(x=0\).
The method relies on understanding Maclaurin Series (8.2) and Manipulating Maclaurin Series (8.3).
In order to find the required derivatives you will probably have to use Implicit Differentiation (4.10) and the Product Rule (4.8).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?