Welcome to First Order Differential Equations!

Hello mathematicians! This chapter is where Pure Mathematics really starts tackling real-world problems. Differential equations (or ODEs) look intimidating, but they are simply equations involving a function and its derivatives.

Our goal in this chapter is not just to differentiate, but to integrate backwards. We will be given the rate of change (\(\frac{dy}{dx}\)) and we have to find the original function \(y(x)\).

Think of a differential equation as a set of rules governing how a quantity grows or decays (like population, temperature, or current). By solving it, we uncover the exact formula for that quantity over time.

What is a First Order Differential Equation?

  • Order: The order of an ODE is determined by the highest derivative present. Since we are in FP2, we focus only on First Order, meaning the highest derivative is \(\frac{dy}{dx}\) or \(\dot{y}\).
  • Example Form: \(\frac{dy}{dx} = f(x, y)\) or \(a(x)\frac{dy}{dx} + b(x)y = c(x)\).

I. Method 1: Separable Variables

This is the nicest type of ODE. If you can use this method, always do so, as it is the fastest!

What Makes an ODE Separable?

A first-order differential equation is separable if you can rearrange it so that all terms involving \(y\) (and \(dy\)) are on one side, and all terms involving \(x\) (and \(dx\)) are on the other side.

The standard form is:
$$\frac{dy}{dx} = f(x)g(y)$$

Step-by-Step Guide for Separation
  1. Separate: Rewrite the equation so you have \(g(y)\) with \(dy\) and \(f(x)\) with \(dx\).
    $$\frac{1}{g(y)} dy = f(x) dx$$
  2. Integrate: Integrate both sides with respect to their respective variables.
    $$\int \frac{1}{g(y)} dy = \int f(x) dx$$
  3. Add Constant: Add the arbitrary constant \(+C\) to one side only (usually the \(x\)-side).
  4. Solve: If required, solve the resulting implicit equation for \(y\).

!!! Common Mistake Alert !!!
Never forget the constant of integration, \(+C\)! If you forget \(C\), you will only find a specific solution, not the required General Solution (a loss of marks!).

Key Takeaway: If you can 'split' the variables cleanly, integrate both sides immediately.

II. Method 2: Homogeneous Equations

Don't worry if this sounds scary—"homogeneous" just means "of the same type." We use a clever trick (a substitution) to turn this complicated equation into a simple separable one we already know how to solve!

Identifying a Homogeneous ODE

An ODE \(\frac{dy}{dx} = f(x, y)\) is homogeneous if the function \(f(x, y)\) can be written purely in terms of the ratio \(\frac{y}{x}\).

Example: If you see terms like \(x^2 + y^2\) divided by \(xy\), notice that if you divide the numerator and denominator by \(x^2\), you get \(\frac{1 + (y/x)^2}{y/x}\). This is homogeneous!

The Homogeneous Substitution Trick

The core substitution for this method is always:
$$\mathbf{y = vx}$$
where \(v\) is a new function of \(x\).

The Critical Derivative Step

If \(y = vx\), we must find \(\frac{dy}{dx}\) using the Product Rule:
$$\frac{dy}{dx} = (v)(1) + (x)\left(\frac{dv}{dx}\right)$$
$$\mathbf{\frac{dy}{dx} = v + x \frac{dv}{dx}}$$

Mnemonic: Remember to substitute both \(y\) and \(\frac{dy}{dx}\) into the original equation.

Step-by-Step Guide for Homogeneous Equations
  1. Identify: Check if the ODE is homogeneous (can be written using only \(\frac{y}{x}\)).
  2. Substitute: Replace \(y\) with \(vx\) and \(\frac{dy}{dx}\) with \(v + x \frac{dv}{dx}\).
  3. Separate: After substitution, the resulting equation will involve only \(v\) and \(x\). You must rearrange it into a separable form (all \(v\) terms with \(dv\), all \(x\) terms with \(dx\)).
  4. Integrate: Integrate both sides and find a solution relating \(v\) and \(x\).
  5. Replace \(v\): Finally, substitute back \(v = \frac{y}{x}\) to get the solution in terms of \(x\) and \(y\).

Did you know? Homogeneous ODEs are often used in trajectory and geometric problems because scaling the coordinates \(x\) and \(y\) by the same factor \(t\) doesn't change the slope \(\frac{dy}{dx}\).

Key Takeaway: Homogeneous means use the substitution \(y=vx\). This turns a tricky equation into a familiar separable one.

III. Method 3: Linear First-Order Equations (The Integrating Factor)

The Integrating Factor (I.F.) method is used for linear first-order ODEs. This method is incredibly powerful but requires strict adherence to the rules.

Identifying a Linear ODE

An ODE is linear if it can be written in the specific Standard Form:
$$\mathbf{\frac{dy}{dx} + P(x)y = Q(x)}$$

Note: \(P(x)\) and \(Q(x)\) must be functions of \(x\) only (or constants). The function \(y\) and its derivative \(\frac{dy}{dx}\) must appear only to the power of one (i.e., no \(y^2\) or \(y\frac{dy}{dx}\) terms).

Pre-requisite check: If your equation isn't in this form, divide or multiply until the coefficient of \(\frac{dy}{dx}\) is exactly 1.

The Integrating Factor (I.F.) Concept

The problem with the standard form is that the Left Hand Side (LHS) is not easily integrable. The I.F. is a "magic multiplier" that, when multiplied across the entire equation, forces the LHS to become the result of a Product Rule differentiation: \(\frac{d}{dx}[y \cdot I(x)]\).

The Integrating Factor, \(I(x)\), is defined as:
$$\mathbf{I(x) = e^{\int P(x) dx}}$$

Don't worry about the \(+C\) when calculating \(\int P(x) dx\). We only need *one* valid I.F., and the constant \(C\) will appear later.

Step-by-Step Guide for the Integrating Factor Method
  1. Standard Form: Ensure the ODE is in the form \(\frac{dy}{dx} + P(x)y = Q(x)\). Identify \(P(x)\) and \(Q(x)\).
  2. Calculate I.F.: Find the integrating factor \(I(x) = e^{\int P(x) dx}\).
  3. Multiply: Multiply the entire standard form equation by the I.F., \(I(x)\).
  4. Simplify LHS: The LHS will automatically condense into the product rule form.
    $$I(x) \left( \frac{dy}{dx} + P(x)y \right) = \frac{d}{dx} [y \cdot I(x)]$$
  5. Integrate: Integrate both sides with respect to \(x\).
    $$\int \frac{d}{dx} [y \cdot I(x)] dx = \int Q(x) I(x) dx$$
    $$y \cdot I(x) = \int Q(x) I(x) dx \mathbf{+ C}$$
  6. Solve for \(y\): Divide by \(I(x)\) to isolate \(y\).

Quick Review Box: The I.F. Mnemonic (SPIIS)

  • Standard Form: \(\frac{dy}{dx} + P(x)y = Q(x)\).
  • P: Identify \(P(x)\).
  • I: Calculate the I.F. \(e^{\int P(x) dx}\).
  • I: Integrate: \(y \cdot I(x) = \int Q(x) I(x) dx + C\).
  • S: Solve for \(y\).

Key Takeaway: The I.F. is essential for solving linear ODEs. It converts the difficult LHS into the result of a derivative, making it easy to reverse the differentiation process.

IV. General and Particular Solutions

When you solve a first-order ODE, your solution will always fall into one of two categories, depending on whether you have information about boundary conditions.

1. The General Solution

The solution that includes the arbitrary constant \(C\) (or \(\ln A\), etc.).

  • It represents a family of curves.
  • If you sketch this solution for different values of \(C\), you get many curves, all satisfying the original rate of change relationship.

2. The Particular Solution

If the question provides a Boundary Condition (B.C.)—a specific point \((x_0, y_0)\) the function must pass through—you can determine the exact value of \(C\).

The resulting solution, without the arbitrary constant \(C\), is called the Particular Solution.

How to Find the Particular Solution
  1. Find the General Solution: Solve the ODE using one of the methods above (Separable, Homogeneous, or I.F.), keeping \(+C\).
  2. Substitute B.C.: Plug in the given \(x\) and \(y\) values into the General Solution.
  3. Calculate \(C\): Solve the resulting numerical equation for \(C\).
  4. State the Particular Solution: Substitute the numerical value of \(C\) back into the General Solution.

Encouragement: You are essentially using the General Solution (the 'rule') and the Boundary Condition (the 'starting point') to find the exact path of the curve!

V. Summary of Techniques

When you see a first-order ODE, use this flowchart to decide which method to apply:

If the ODE... Standard Form / Test Method to Use
Can be split into \(f(x) dx = g(y) dy\) \(\frac{dy}{dx} = f(x)g(y)\) Separable Variables (Direct Integration)
Can be written entirely in terms of \(\frac{y}{x}\) \(\frac{dy}{dx} = F\left(\frac{y}{x}\right)\) Homogeneous (Substitution: \(y = vx\))
Fits the linear format exactly \(\frac{dy}{dx} + P(x)y = Q(x)\) Integrating Factor (\(I(x) = e^{\int P(x) dx}\))

Final Tip for Success: Always check if you can simplify or separate before jumping straight into the Integrating Factor method, as separation is usually faster! And remember, practice is key—the more you recognize the form, the easier the solution method becomes. You've got this!