# First Order Differential Equations (FP2)

> Edexcel International A-Level Further Mathematics · FP2
> Source: https://www.owlsprep.com/study/edexcel-ial-further-math-fp2-first-order-differential-equations/

This guide covers all three first-order differential equation techniques required for Edexcel IAL Further Maths FP2: separable ODEs, linear ODEs solved via integrating factor, and ODEs reducible using given substitutions, plus exam-focused problem-solving strategies.

**Prerequisites:** [A-Level Pure Mathematics integration techniques](https://www.owlsprep.com/study/edexcel-ial-pure-maths-integration/); [Basic differentiation rules (chain, product, quotient)](https://www.owlsprep.com/study/edexcel-ial-pure-maths-differentiation/)

## Learning objectives

- Solve separable first-order ODEs, find particular solutions using boundary conditions, and sketch families of solution curves
- Use the integrating factor method to solve linear first-order ODEs of the form $\frac{dy}{dx} + Py = Q$
- Apply given substitutions to reduce non-standard first-order ODEs to separable or linear forms
- Form first-order ODEs from real-world contexts as required by the exam specification

## Separable First Order ODEs

**Separable First Order ODE** — A differential equation of the form $\frac{dy}{dx} = f(x)g(y)$, which can be rearranged to separate variables on opposite sides of the equation: $\frac{1}{g(y)} dy = f(x) dx$.

To solve, integrate both sides of the rearranged equation, adding a single arbitrary constant $C$ to the x-side to get the general solution. If boundary conditions are provided, substitute them to solve for $C$ and get a particular solution.

**Worked example:** Find the particular solution of $\frac{dy}{dx} = 3x^2 y$ given that $y=2$ when $x=0$.

1. 1. Rearrange to separate variables:
2. $$\frac{1}{y} dy = 3x^2 dx$$
3. 2. Integrate both sides:
4. $$\ln|y| = x^3 + C$$
5. 3. Exponentiate both sides to eliminate the natural log:
6. $$y = Ae^{x^3} \text{ where } A = e^C$$
7. 4. Apply the boundary condition $y=2$, $x=0$:
8. $$2 = Ae^{0} \implies A = 2$$
9. 5. Write the particular solution:
10. $$y = 2e^{x^3}$$

> **Exam tip:** Always include the modulus sign when integrating $1/y$ to get $\ln|y|$, unless the context explicitly states $y$ is always positive; you can drop it later when combining constants if appropriate.

*Calculator:* allowed

## Linear First Order ODEs & Integrating Factor

**Linear First Order ODE** — A differential equation that can be written in the standard form $\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x)$ and $Q(x)$ are functions of $x$ only.

The integrating factor (IF) method is used to solve these equations. The IF is calculated as $e^{\int P(x) dx}$; you do not need to add a constant of integration when calculating the IF. Multiply every term in the standard form equation by the IF, and the left-hand side will simplify to the derivative of the product of IF and $y$.

**Worked example:** Solve $\frac{dy}{dx} + \frac{2}{x}y = x$ for $x>0$.

1. 1. Identify $P(x)$ and $Q(x)$ from the standard form: $P(x) = 2/x$, $Q(x) = x$.
2. 2. Calculate the integrating factor:
3. $$IF = e^{\int \frac{2}{x} dx} = e^{2 \ln x} = e^{\ln x^2} = x^2$$
4. 3. Multiply all terms in the original equation by IF:
5. $$x^2 \frac{dy}{dx} + 2x y = x^3$$
6. 4. Recognize the left-hand side is the derivative of $IF \cdot y$:
7. $$\frac{d}{dx}(x^2 y) = x^3$$
8. 5. Integrate both sides with respect to $x$:
9. $$x^2 y = \frac{x^4}{4} + C$$
10. 6. Rearrange for $y$ to get the general solution:
11. $$y = \frac{x^2}{4} + \frac{C}{x^2}$$

> **tip**
>
> Always rearrange the ODE into the exact standard form $\frac{dy}{dx} + P(x)y = Q(x)$ before identifying $P(x)$: if the coefficient of $\frac{dy}{dx}$ is not 1, divide through by that coefficient first.

*Calculator:* allowed

## ODEs Reducible via Given Substitutions

Some non-standard first-order ODEs cannot be solved directly with separable or integrating factor methods, but the exam will always give you a substitution to convert the equation into one of these standard forms. You will need to differentiate the substitution to replace $\frac{dy}{dx}$ with an expression in the new variable and $x$, then substitute into the original ODE.

**Worked example:** Use the substitution $z = y/x$ to reduce $\frac{dy}{dx} = \frac{y^2 + x^2}{xy}$ for $x>0$, $y>0$ to a separable ODE, then find its general solution.

1. 1. Rearrange the given substitution to express $y$ in terms of $z$ and $x$: $y = zx$.
2. 2. Differentiate $y$ with respect to $x$ using the product rule:
3. $$\frac{dy}{dx} = z + x \frac{dz}{dx}$$
4. 3. Substitute $y = zx$ and $\frac{dy}{dx} = z + x \frac{dz}{dx}$ into the original ODE:
5. $$z + x \frac{dz}{dx} = \frac{(zx)^2 + x^2}{x(zx)} = \frac{z^2 x^2 + x^2}{z x^2} = \frac{z^2 + 1}{z} = z + \frac{1}{z}$$
6. 4. Simplify to eliminate $z$ from both sides, then separate variables:
7. $$x \frac{dz}{dx} = \frac{1}{z} \implies z dz = \frac{1}{x} dx$$
8. 5. Integrate both sides:
9. $$\frac{z^2}{2} = \ln x + C$$
10. 6. Substitute back $z = y/x$ to get the general solution in terms of $x$ and $y$:
11. $$\frac{y^2}{2x^2} = \ln x + C \implies y^2 = 2x^2 (\ln x + C)$$

> **Exam tip:** Always substitute back to the original variables at the end of your solution; the exam will almost always ask for your answer in terms of $x$ and $y$, not the substituted variable.

*Calculator:* allowed

## Forming ODEs and Sketching Solution Curves

Exam questions may require you to form a first-order ODE from a context (e.g. rates of change, population growth, cooling) using given proportional relationships. You may also be asked to sketch members of a family of solution curves, showing the effect of different values of the arbitrary constant $C$.

**Worked example:** The rate of increase of a population $P$ at time $t$ is proportional to the product of $P$ and $(1000 - P)$. Form a first-order ODE for $P$ in terms of $t$, using $k$ as the constant of proportionality.

1. 1. Translate the rate description into mathematical terms: the rate of increase of $P$ is $\frac{dP}{dt}$.
2. 2. State the proportional relationship: $\frac{dP}{dt} \propto P(1000 - P)$
3. 3. Add the constant of proportionality $k$ to get the final ODE:
4. $$\frac{dP}{dt} = kP(1000 - P) \text{ where } k > 0$$

When sketching solution curves, plot 2-3 curves for different values of $C$ (e.g. positive, negative, zero), label any asymptotes or intercepts, and clearly note how changing $C$ shifts or transforms the curve.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Forgetting to rearrange linear ODEs to standard form (coefficient of $\frac{dy}{dx} = 1$) before calculating the integrating factor.
  - Why it fails: This leads to an incorrect $P(x)$ value, so the IF is wrong and the entire solution fails.
  - Correct: Always divide through by the coefficient of $\frac{dy}{dx}$ first to make it 1, then identify $P(x)$ and $Q(x)$.
- **Wrong:** Adding a constant of integration when calculating the integrating factor.
  - Why it fails: The constant would cancel out when multiplied through the equation, so it is unnecessary and wastes time, and can lead to errors if misapplied.
  - Correct: Omit the constant of integration when computing the integral for the IF.
- **Wrong:** Dropping the modulus sign when integrating $1/y$ to get $\ln(y)$ without justification.
  - Why it fails: This can lead to missing negative solutions if $y$ can be negative, and examiners may deduct marks for missing the modulus.
  - Correct: Write $\ln|y|$ first, then drop the modulus only when combining with the arbitrary constant or if the context specifies $y$ is always positive.
- **Wrong:** Forgetting to substitute back to the original variables after using a given substitution.
  - Why it fails: The question almost always requires the answer in terms of the original $x$ and $y$ variables, so leaving it in terms of the substituted variable will lose marks.
  - Correct: Always replace the substituted variable with the original expression as the final step of your solution.
- **Wrong:** Adding separate arbitrary constants to both sides when integrating separable ODEs.
  - Why it fails: Two separate constants can be combined into a single constant, so adding both is redundant and can lead to confusion when applying boundary conditions.
  - Correct: Add a single arbitrary constant $C$ to the right-hand (x) side only after integrating both sides.

## Cheatsheet

| Method | Standard Form | Key Steps |
| --- | --- | --- |
| Separable ODE | $\frac{dy}{dx} = f(x)g(y)$ | 1. Rearrange to $\frac{1}{g(y)} dy = f(x) dx$; 2. Integrate + C; 3. Apply boundary conditions for particular solution. |
| Integrating Factor | $\frac{dy}{dx} + P(x)y = Q(x)$ | 1. Calculate $IF = e^{\int P(x) dx}$; 2. Multiply equation by IF; 3. Simplify LHS to $\frac{d}{dx}(IF \cdot y)$; 4. Integrate both sides. |
| Given Substitution | Any non-standard first-order ODE | 1. Differentiate given substitution to replace $\frac{dy}{dx}$; 2. Substitute into original ODE; 3. Solve resulting standard ODE; 4. Substitute back to original variables. |

## What's next

Mastering first-order differential equations is a foundational skill for the rest of your FP2 studies, and it is frequently combined with other topics like integration and polar coordinates in exam questions. Once you are confident solving the three types of first-order ODEs covered here, your next step is to move on to second-order linear differential equations, the next major ODE topic in FP2. You should also practice linking ODE formation to real-world contexts, as these questions often appear alongside mechanics or statistics topics in your further maths exams. Regular practice of past paper questions on this topic will help you avoid common mistakes and speed up your problem-solving under timed exam conditions.

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