Unit 8 - Advanced Calculus (HL only)
Required Prior Knowledge
Questions
Solve$$\int x\sin x dx$$
Solutions
Get Ready
Questions
Consider the differential equation$$\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)\qquad\qquad (1)$$Show that this is NOT separable.
Now consider this$$I=e^{\int P\left(x\right)dx}\qquad\qquad (2)$$Find \(\frac{dI}{dx}\). Label this equation \((3)\).
Find the derivative of \(Iy\) with respect to \(x\). Label this \((4)\) (Hint: use implicit differentiation with the product rule).
By multiplying \((1)\) by \(I\) we get:$$I\frac{dy}{dx}+IP\left(x\right)y=IQ\left(x\right)\qquad\qquad (5)$$
By comparing \((5)\) with \((4)\):$$\frac{d}{dx}\left(Iy\right)=IQ\left(x\right)\qquad\qquad (6)$$And so$$Iy=$$
Solutions
Notes
For a differential equation of the form$$\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$$we can find a solution by mutliplying by an integrating factor using these steps:
Find the integrating factor$$I\left(x\right)=e^{\int P\left(x\right)dx}$$This is given in the formula booklet. You can ignore the constant of integration at this stage.
Set up the equation$$I\left(x\right)y=\int I\left(x\right)Q\left(x\right)dx$$.
Integrate the right hand side, and rearrange to obtain the general solution.
Examples and Your Turns
Example
Find the general solution of$$\frac{dy}{dx}+\frac{2}{x}y=5x^{2}$$
Your Turn
Find the general solution of$$\frac{dy}{dx}+\frac{1}{x}y=3x$$
Your Turn
Find the general solution of$$\frac{dy}{dx}+2y=e^{x}$$
Your Turn
Find the general solution of$$\frac{dy}{dx}+3x^{2}y=6x^{2}$$
Your Turn
Find the general solution of$$\frac{dy}{dx}+y\tan x=\cos^{2}x$$
Your Turn
Find the general solution of$$\left(x+1\right)\frac{dy}{dx}-3y=\left(x+1\right)^{5}$$
Your Turn
Find the solution of the differential equation$$\frac{dy}{dx}-3y=3$$given that \(y\left(0\right)=2\).
Your Turn
Find the solution of the differential equation$$\frac{dy}{dx}-y\tan x=-\sec x$$given that \(y\left(0\right)=1\).
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
Why does multiplying by the integrating factor always work for differential equations of this form? Look back at the Get Ready.
What happens if \(Q\left(x\right)=0\), and how does this link to other methods we have seen to solve differential equations?
Common Mistakes / Misconceptions
Not paying close attention to the form of the equation. In \(\frac{dy}{dx}-2y=x\), we have that \(P\left(x\right)=-2\), but many students forget the minus sign or even the \(2\).
The most common mistake is to not recall the second part of the method: you need to know that \(Iy=\int I Q\left(x\right)dx\).
Be careful to ensure you fully simplify the integrating factor to make the rest of the question easier on yourself.
Errors in the integration are also common - you need to know your integration methods!
Connecting This to Other Skills
This is another method of solving Differential Equations (8.4) when they are not Separable (8.6) or Homogenous (8.7).
We have used the Product Rule (4.8) and Implicit Differentiation (4.10) to derive the integrating factor.
The final integrations you will have to do will require knowledge of Indefinite Integration (5.2), Reverse Chain Rule (5.4), Integration by Substitution (5.7) and commonly Integration By Parts (5.8).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?