Welcome to the World of Change: Differential Equations!
Ever wondered how scientists predict how fast a population of rabbits will grow, or how quickly a hot cup of tea cools down? They use Differential Equations (DEs)! While normal equations like \(x + 2 = 5\) help us find a specific number, differential equations help us find a function (a relationship between variables) by looking at how things change.
In this chapter, we are going to learn how to "reverse-engineer" these rates of change to find the original formula. Don't worry if it seems a bit abstract at first—think of it like being a math detective, looking at the tracks (the derivative) to find out what animal (the function) made them!
Prerequisite Check: Before we start, make sure you are comfortable with basic Integration and Differentiation. If you can integrate \( \frac{1}{x} \) and \( e^x \), you’re already halfway there!
1. Solving Separable Differential Equations
The most common type of DE you will meet in the H2 syllabus looks like this:
\( \frac{dy}{dx} = f(x)g(y) \)
This just means the derivative is a mix of \(x\) stuff and \(y\) stuff multiplied together. To solve this, we use a technique called Separation of Variables.
How to Separate Variables (The "Social Distancing" Method)
Think of \(x\) and \(y\) as two groups that need to stay on their own sides of the equal sign. Your goal is to get all the \(y\) terms with \(dy\) on the left, and all the \(x\) terms with \(dx\) on the right.
Step-by-Step Process:
1. Rearrange: Move the \(g(y)\) term to the left (it usually becomes \( \frac{1}{g(y)} \)) and the \(dx\) to the right.
\( \frac{1}{g(y)} dy = f(x) dx \)
2. Integrate: Add integral signs to both sides.
\( \int \frac{1}{g(y)} dy = \int f(x) dx \)
3. Solve: Perform the integration on both sides.
4. Add \(C\): Don’t forget the constant of integration! (Usually, we just add it to the \(x\) side).
5. Simplify: If possible, rearrange the final equation to make \(y\) the subject (the explicit form).
Example: Solve \( \frac{dy}{dx} = \frac{x}{y} \)
1. Separate: \( y \ dy = x \ dx \)
2. Integrate: \( \int y \ dy = \int x \ dx \)
3. Solve: \( \frac{y^2}{2} = \frac{x^2}{2} + C \)
4. Simplify: \( y^2 = x^2 + 2C \) (or \( y^2 = x^2 + A \) where \(A\) is a constant).
Quick Review: Separation Success
Key Takeaway: If you can't get all \(y\)'s on one side and all \(x\)'s on the other through multiplication or division, it’s not a separable DE!
2. General vs. Particular Solutions
When you solve a DE, you usually end up with a constant \(C\). This is called the General Solution because it represents a whole family of curves.
However, if the question gives you a specific point (e.g., "when \(x = 0, y = 1\)"), you can find the exact value of \(C\). This is called the Particular Solution.
Analogy:
- General Solution: "I am taking a bus to a city." (Which bus? Which city? We don't know yet!)
- Particular Solution: "I am taking Bus 168 to Bedok." (Everything is specific.)
Common Mistake to Avoid:
Always add your constant \(+C\) immediately after integrating. Do not wait until the very end of the algebra to add it, or your final answer will be wrong!
3. Reducing DEs Using Substitution
Sometimes, a DE looks messy and isn't separable. In these cases, the question will provide you with a substitution (like \(v = y - x\) or \(y = vx\)) to make it easier.
How to handle substitutions:
1. Differentiate the substitution: If you are given \(v = y - x\), find \( \frac{dv}{dx} \).
2. Replace everything: Substitute your new expressions for \(y\) and \( \frac{dy}{dx} \) into the original DE.
3. Solve the new DE: It should now be separable in terms of \(v\) and \(x\).
4. Substitute back: Once you've solved it, replace \(v\) with the original \(y\) expression.
Don't worry if this seems tricky at first! Just remember: the goal of substitution is to "mask" the complicated parts so you can use your Separation of Variables skills.
4. Formulating DEs (Modeling)
This is where we apply DEs to real life. You will be given a word problem and asked to "formulate a differential equation."
Decoding the "Math Language":
- "Rate of increase of \(N\)" means \( \frac{dN}{dt} \).
- "Is proportional to..." means \( = k \times (\dots) \).
- "Rate of decrease" means you must include a negative sign (\( -k \)).
Example Situation:
"The rate at which a population \(P\) grows is proportional to the square root of the population."
Translated to Math: \( \frac{dP}{dt} = k\sqrt{P} \)
Did you know? This is exactly how radioactive decay works! The rate at which an element disappears is proportional to how much of it is left: \( \frac{dm}{dt} = -km \).
Key Takeaway for Modeling
Always identify the independent variable (usually time, \(t\)) and the dependent variable (the thing that is changing). Watch out for words like "inversely proportional," which means you divide by the variable instead of multiplying.
5. Interpreting the Solution
After finding a solution, you might be asked to describe what happens in the "long run."
Look for the Horizontal Asymptote:
As \(t \to \infty\), what happens to \(y\)?
For example, if your solution is \( y = 100 - 50e^{-t} \), as \(t\) gets very large, \(e^{-t}\) becomes \(0\). So, \(y\) approaches \(100\). This might represent the maximum capacity of a lake or the terminal velocity of a falling object.
Summary Checklist
1. Can I separate the variables? (Get \(y\) on the left, \(x\) on the right).
2. Did I remember \(+C\)? (Add it right after integration!).
3. Did I use the initial conditions? (Plug in \(x\) and \(y\) to find the particular solution).
4. Can I translate words to rates? (Rate = derivative).
5. Is my algebra sound? (Be careful with logarithms and exponents when simplifying!).
You've got this! Differential equations are just the "final boss" of integration. Master the separation technique, and the rest is just careful calculation.