Unit 8 - Advanced Calculus (HL only)
Required Prior Knowledge
Questions
Find the general solution to the differential equation $$x\frac{dv}{dx}=\frac{1-v^{2}}{v}$$
Solutions
Get Ready
Questions
Show that
a) $$\frac{x^2 - xy + y^2}{x^2} = 1 - \frac{y}{x} + \left(\frac{y}{x}\right)^2$$
b) $$\frac{x^2y + y^3}{x^3} = \frac{y}{x} + \left(\frac{y}{x}\right)^3$$
c) $$\frac{x^2 + y^2}{xy} = \frac{1 + \left(\frac{y}{x}\right)^2}{\frac{y}{x}}$$
d) $$\frac{x^3 + 2y^3}{x^2y - 3x^3} = \frac{1 + 2\left(\frac{y}{x}\right)^3}{\frac{y}{x} - 3}$$
Solutions
Notes
A homogenous differential equation is one of the form $$\frac{dy}{dx}=f\left(\frac{y}{x}\right)$$
That is, we can rewrite it with \(\frac{y}{x}\) as the argument of the function.
Consider the examples for the Get Ready to see some ways this can be done.
Homogenous Differential Equations can be solved by making the substitution \(y=vx\), which will always produce a separable differential equation.
By differentiating \(y=vx\) we get$$\frac{dy}{dx}=v+x\frac{dv}{dx}$$using implicit differentiation and the product rule.
We then substitute both \(y\) and \(\frac{dy}{dx}\) into the differential equation to get a new equation in terms of \(v\) and \(x\).
Examples and Your Turns
Example
Show that $$\frac{dy}{dx}=\frac{x+2y}{x}$$is homogenous and hence solve the differential equation if \(y=\frac{3}{2}\) when \(x=3\).
Your Turn
Find the general solutions of $$x^{2}\frac{dy}{dx}=x^{2}-xy+y^{2}$$by first showing it is homogenous.
Your Turn
Find the general solution to $$x\frac{dy}{dx}-y=x^{2}\sin x$$
Your Turn
Solve the differential equation $$xy’=y\left(\ln x - \ln y\right)$$when \(y\left(1\right)=4\).
Your Turn
Find the general solution to the differential equation$$\frac{dy}{dx}=\frac{xy+y^{2}}{x^{2}}$$
Your Turn
Find the general solution to the differential equation$$\frac{dy}{dx}=\cos^{2}\left(\frac{y}{x}\right)+\frac{y}{x}$$
Your Turn
Find the general solution to the differential equation$$\frac{dy}{dx}=\frac{y^{2}-x^{2}}{2xy}$$
Your Turn
Suppose$$\frac{dy}{dx}=\frac{x+y}{x-y}$$and \(y\left(1\right)=1\). Find the exact value of \(x\), when \(y=0\).
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 the substitution \(y=vx\) reduce a homogenous differential equation to a separable differential equation?
Common Mistakes / Misconceptions
The first common issue with this type of question is spotting that it is a homogenous differential equation. You are not normally given this hint, so you have to be looking for it.
The second common mistake is forgetting the substitution \(y=vx\) as this is NOT in the formula booklet.
If you remember this step, then the next misconception is to not substitute for the derivative \(\frac{dy}{dx}=v+x\frac{dv}{dx}\), and only substitute for \(y\).
Remember to substitute back in at the end to get a result for \(y\) not \(v\).
After this, the misconceptions are the same as for a separable differential equation.
Connecting This to Other Skills
You require the Product Rule (4.8) and Implicit Differentiation (4.10) to be able to calculate \(\frac{dy}{dx}\) for the substitution.
You also need to be secure in the techniques of Integration (Unit 5) and solving Separable Differential Equations (8.6).
If a differential equation is not separable or homogenous, there is one more method we will see next: 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?