Unit 8 - Advanced Calculus (HL only)
Required Prior Knowledge
Questions
a) Given that \(y=\ln\left(x^{2}+1\right) \) find \(\frac{dy}{dx}\)
b) If \(\frac{dy}{dx}=4x^{3}-2\) determine the function
c) Verify that \(x=2\) is a solution to the equation \(x^{2}+x-6=0\)
d) Verify that \(x=2, y=-3\) is a solution to the equation \(5x+2y=4\)
Solutions
Get Ready
Questions
The population \(P\) of rodents on an island is currently \(500\). Its growth rate is modelled by:$$\frac{dP}{dt}=0.1P\left(1-\frac{P}{3000}\right)$$where \(t\) is the time in years from now.
a) How is this equation different to a function like \(P\left(t\right)=100e^{0.2t}\)?
b) Can you find the expected population in \(5\) years?
c) Can you find the expected time for the population to increase to \(2000\)?
d) Can you find the maximum population of rodents?
Solutions
Notes
A differential equation (often abbreviated to DE) is an equation involving a derivative.
They connect a rate of change to the variables involved in the problem.
Some examples of differential equations could be:$$\begin{matrix}\frac{dy}{dx}=\frac{y}{x}&\frac{dy}{dx}=8y^{2}&\frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}-y=0\end{matrix}$$The simplest differential equations are things like \(\frac{dy}{dx}=3x^{2}\).
We can solve these using simple integration methods to obtain \(y=x^{3}+c\). This is known as the general solution to the differential equation.
More complex differential equation require different methods to find solutions, but there is always a family of solutions that satisfy a differential equation.
If we want to find a particular solution, then we need an initial condition.
The order of a differential equation is the highest derivative in the equation.
The first two examples above are First Order Differential Equations. The final example is a Second Order Differential Equation.
In this course we study the solution of some 1st Order Differential Equations.
A solution to a differential equation is a function that makes the equation true when you substitute the function and its derivatives into the differential equation.
Just like we can verify a solution for a normal equation by substituting the values into the equation to show that they make the equation true, we can substitute functions into a differential equation.
Steps to verify a function is a solution to a differential equation:
1. Differentiate the function to find \(\frac{dy}{dx}\) (and further derivatives if a higher order DE)
2. Substitute the function and / or derivative into the left hand side of the DE
3. Substitute the function and / or derivative into the right hand side of the DE
4. Show these two are the same.
Examples and Your Turns
Example
Verify that \(y=e^{x^{2}}\) is a solution to the differential equation$$\frac{dy}{dx}=2xy$$
Your Turn
Verify that \(y=-\frac{1}{x}\) is a solution to the differential equation$$\frac{dy}{dx}=y^{2}$$
Your Turn
Consider the differential equation$$\frac{dy}{dx}-3y=3$$
a) Verify that \(y=ce^{3x}-1\) is a solution to the differential equation for any value \(c\).
b) Find the particular solution which passes through \(\left(0,2\right)\).
c) Find the equation of the tangent to the particular solution at \(\left(0,2\right)\).
Your Turn
Consider the differential equation$$\frac{dy}{dx}=\frac{x}{y}$$
a) Verify that \(x^{2}+y^{2}=K\) is a solution to the differential equation for any value \(K\).
b) Find the particular solution which passes through \(\left(3,-1\right)\).
c) State the coordinates of the point on this curve where the tangent is vertical.
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
Why does a general solution represent a family of curves, rather than just one specific curve?
What is the difference between \(\frac{dy}{dx}=x^{2}\) and \(\frac{dy}{dx}=y^{2}\)? Why is the second DE harder to solve?
Common Mistakes / Misconceptions
Forgetting to include the constant term that represents the general solution.
Mistaking the two notations \(\frac{d^{2}y}{dx^{2}}\) and \(\left(\frac{dy}{dx}\right)^{2}\).
Connecting This to Other Skills
You will need all your Differentiation skills to verify a solution, such as Differentiating Polynomials (4.5), the Chain Rule (4.7), Product Rule (4.8), Quotient Rule (4.9), Implicit Differentiation (4.10) and Differentiating Exponentials (4.11) and Trig (4.12).
Finding the Equations of Curves (5.3) is the main conceptual idea underpinning finding particular solutions to differential equations.
We will see the methods to solve differential equations in Euler’s Method (8.5), Separable Differential Equations (8.6), Homogenous Differential Equations (8.7) and Integrating Factor Method (8.8).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?