Introduction to Differential Equations

Welcome to one of the most powerful tools in mathematics! A differential equation (DE) is simply an equation that involves a derivative, like \(\frac{dy}{dx}\). While standard algebra helps us find a specific number, differential equations help us find a function. These equations are the language of the universe—they describe how populations grow, how heat moves through a room, and how planets orbit the sun.

In this chapter, we focus on first-order differential equations. This means the highest derivative in the equation is the first derivative, \(\frac{dy}{dx}\). We will explore three analytical methods to find exact solutions and one numerical method to find an approximation when the "exact" way is too difficult.

1. Variables Separable

This is the most straightforward method. If you can move every term with a \(y\) to the left side and every term with an \(x\) to the right side, the equation is separable.

How to Solve:

1. Rearrange the equation into the form: \(f(y) dy = g(x) dx\).
2. Integrate both sides: \(\int f(y) dy = \int g(x) dx\).
3. Always include a constant of integration, \(+ C\), usually on the \(x\) side.
4. If you are given a specific point \((x, y)\), substitute it to find the value of \(C\). This is called a particular solution.

Quick Review: Remember that \(\frac{dy}{dx}\) is not a fraction, but in this method, we treat it like one to "separate" the \(dx\) and \(dy\). This is a handy trick that works because of the Chain Rule!

Common Mistake: Forgetting the \(+ C\) until the very end. You must add the constant as soon as you integrate. If you add it later, you might get the wrong final function!

2. Homogeneous Equations (Substitution \(y = vx\))

Sometimes, variables aren't easily separated because they are "tangled up" in a specific way. If a differential equation can be written as \(\frac{dy}{dx} = f\left(\frac{y}{x}\right)\), we call it homogeneous.

The Strategy:

We use a substitution to turn a difficult equation into a simple separable one.

1. Let \(y = vx\), where \(v\) is a function of \(x\).
2. Use the product rule to differentiate: \(\frac{dy}{dx} = v + x\frac{dv}{dx}\).
3. Substitute both \(y\) and \(\frac{dy}{dx}\) into your original equation.
4. The \(x\)'s should cancel out, leaving you with a separable equation involving \(v\) and \(x\).
5. Solve for \(v\), and finally, substitute \(v = \frac{y}{x}\) back in to get your answer in terms of \(y\) and \(x\).

Analogy: Think of this like using a "translator." The equation is currently in a language we don't speak (\(y\) and \(x\)), so we translate it into a language we do speak (\(v\) and \(x\)), solve it, and translate it back!

3. The Integrating Factor

This method is used for Linear First-Order Differential Equations that look like this:
\(\frac{dy}{dx} + P(x)y = Q(x)\)

If your equation has a \(y\) and a \(\frac{dy}{dx}\) but they aren't multiplied together, this is your go-to method.

Step-by-Step Process:

1. Standard Form: Ensure the coefficient of \(\frac{dy}{dx}\) is exactly \(1\). If it's not, divide the whole equation by whatever is in front of it.
2. Find the Integrating Factor: Calculate \(I(x) = e^{\int P(x) dx}\). (Ignore the \(+ C\) for this integral).
3. Multiply: Multiply every single term in the differential equation by \(I(x)\).
4. The Magic Step: The left side of your equation now simplifies perfectly to the derivative of a product: \(\frac{d}{dx}(I(x) \cdot y)\).
5. Integrate: Integrate both sides: \(I(x) \cdot y = \int I(x) \cdot Q(x) dx\).
6. Solve for \(y\): Divide by \(I(x)\) to isolate \(y\).

Did you know? The "Integrating Factor" is essentially a "missing piece" that turns the left side of the equation into the result of a Product Rule differentiation.

4. Euler’s Method (Numerical Solution)

Sometimes, an equation is impossible to solve analytically (we can't find a "nice" formula). In these cases, we use Euler’s Method to estimate the values of the function step-by-step.

The Logic:

If we know a starting point \((x_0, y_0)\) and we know the slope \(\frac{dy}{dx} = f(x, y)\), we can "walk" a small distance \(h\) in that direction to find the next point.

The Formula:

\(x_{n+1} = x_n + h\)
\(y_{n+1} = y_n + h \times f(x_n, y_n)\)

Where:
\(h\) is the step size.
\(f(x_n, y_n)\) is the gradient (slope) at the current point.

Step-by-step Guide:
1. Start with your given initial condition \((x_0, y_0)\).
2. Calculate the gradient at that point using the DE.
3. Multiply the gradient by the step size \(h\) to find the change in \(y\).
4. Add the change in \(y\) to your current \(y\) to get the next \(y\).
5. Repeat until you reach the desired \(x\) value.

Important Note: Smaller values of \(h\) lead to more accurate approximations, but require more steps. Because this involves repetitive calculations, it is often tested with a table in Paper 2 or 3 where a GDC (Graphic Display Calculator) is allowed.

Common Mistake: Mixing up \(x\) and \(y\) in the gradient formula. Double-check your substitutions at every step!

Key Takeaways Summary

  • Separable: Use when you can "split" the variables. \(\int f(y) dy = \int g(x) dx\).
  • Homogeneous: Use the substitution \(y = vx\) if the equation depends on the ratio \(\frac{y}{x}\).
  • Integrating Factor: Use for \(\frac{dy}{dx} + P(x)y = Q(x)\). The factor is \(I(x) = e^{\int P(x) dx}\).
  • Euler's Method: A numerical "stepping" method. \(y_{new} = y_{old} + h \times (\text{slope})\).
  • Boundary Conditions: Use given points to find the specific constant \(C\).

Final Tip: When you see a differential equation on an exam, first check if it's separable. If not, check if it's in the linear form for an Integrating Factor. If it's messy with \(x^2\) and \(xy\) terms, check if it's homogeneous!