# Differential Equations

> CIE A-Level Further Mathematics · Further Pure 2
> Source: https://www.owlsprep.com/study/cie-9231-u2-differential-equations/

This module covers core methods for solving first-order linear differential equations and second-order linear differential equations with constant coefficients, a key exam-weighted topic for CIE A-Level Further Maths FP2.

**Prerequisites:** [Basic differentiation and integration](https://www.owlsprep.com/study/cie-9231-u1-integration-techniques/); [Quadratic equations and roots](https://www.owlsprep.com/study/alevel-pure-quadratic-equations/)

## Learning objectives

- Solve first-order linear differential equations using the integrating factor method
- Solve homogeneous and non-homogeneous second-order linear differential equations with constant coefficients
- Apply boundary conditions to find particular solutions of differential equations

## First-Order Linear Differential Equations

**First-order linear differential equation** — A first-order equation where $y$ and its derivative are both to the first power, with coefficients that can be functions of the independent variable $x$.

*Notation:* \frac{dy}{dx} + P(x)y = Q(x)

*Example:* \frac{dy}{dx} + 3y = e^x

To solve an equation in standard form above, we first calculate the integrating factor $I(x) = e^{\int P(x) dx}$. Multiply both sides of the equation by $I(x)$: the left-hand side simplifies to $\frac{d}{dx}\left(I(x)y\right)$ by the product rule. Integrate both sides to solve for $y$.

**Worked example:** Solve $\frac{dy}{dx} + \frac{y}{x} = x^2$ for $x>0$, given $y=2$ when $x=1$.

1. 1. Identify $P(x) = \frac{1}{x}$ and $Q(x) = x^2$ from standard form.
2. 2. Calculate the integrating factor:
3. $$\int P(x) dx = \int \frac{1}{x} dx = \ln x \implies I(x) = e^{\ln x} = x$$
4. 3. Multiply both sides of the equation by $I(x)$:
5. $$x \frac{dy}{dx} + y = x^3$$
6. 4. Rewrite left-hand side as product rule derivative:
7. $$\frac{d}{dx}\left(xy\right) = x^3$$
8. 5. Integrate both sides, adding constant of integration:
9. $$xy = \frac{x^4}{4} + C$$
10. 6. Apply boundary condition $y=2, x=1$ to find $C$:
11. $$(1)(2) = \frac{1^4}{4} + C \implies C = 2 - \frac{1}{4} = \frac{7}{4}$$
12. 7. Rearrange to get the final solution:
13. $$y = \frac{x^3}{4} + \frac{7}{4x}$$

> **Exam tip:** Always rearrange your equation to get a coefficient of 1 for $\frac{dy}{dx}$ before calculating the integrating factor, to avoid errors in the sign or value of $P(x)$.

## Homogeneous Second-Order Differential Equations

**Homogeneous second-order linear DE (constant coefficients)** — Second-order equation where all terms involve $y$ or its derivatives, with constant coefficients $a,b,c$ and zero right-hand side.

*Notation:* a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c y = 0 \quad (a \neq 0)

We solve these by forming the auxiliary (characteristic) quadratic equation $am^2 + bm + c = 0$. The form of the general solution depends entirely on the nature of the roots of this quadratic:

1. Two distinct real roots $m_1, m_2$: $y = A e^{m_1 x} + B e^{m_2 x}$
2. One repeated real root $m$: $y = (A + Bx) e^{m x}$
3. Two complex conjugate roots $\alpha \pm i\beta$: $y = e^{\alpha x} (A \cos \beta x + B \sin \beta x)$

**Worked example:** Find the general solution of $\frac{d^2 y}{dx^2} - 4 \frac{dy}{dx} + 4 y = 0$.

1. 1. Form the auxiliary equation from the coefficients:
2. $$m^2 - 4m + 4 = 0$$
3. 2. Factorize to find roots:
4. $$(m-2)^2 = 0 \implies m = 2 \text{ (repeated root)}$$
5. 3. Use the general solution formula for repeated roots:
6. $$y = (A + Bx) e^{2x}$$

## Non-Homogeneous Second-Order Differential Equations

**Non-homogeneous second-order linear DE (constant coefficients)** — General solution = Complementary Function (CF, solution of the homogeneous equation) + Particular Integral (PI, any solution that satisfies the full non-homogeneous equation).

*Notation:* a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c y = f(x) \quad (a \neq 0, f(x) \neq 0)

We find the particular integral by the method of undetermined coefficients: guess a form of PI based on $f(x)$, substitute into the equation to solve for unknown coefficients. If your guess matches a term in the complementary function, multiply it by $x$ (or $x^2$ for a repeated root) to get a linearly independent solution.

> **warning**
>
> Failing to adjust your PI guess when it matches the CF will always give an incorrect solution, as substitution will result in 0 = f(x) which cannot be solved.

**Worked example:** Find the general solution of $\frac{d^2 y}{dx^2} - 3 \frac{dy}{dx} + 2 y = 6 e^{4x}$.

1. 1. First find the complementary function by solving the homogeneous equation:
2. $$m^2 - 3m + 2 = 0 \implies (m-1)(m-2) = 0 \implies m = 1, 2$$
3. So the complementary function is:
4. $$CF: y = A e^x + B e^{2x}$$
5. 2. Guess the particular integral: $f(x) = 6e^{4x}$, and 4 is not a root of the auxiliary equation, so try $y = C e^{4x}$.
6. 3. Calculate derivatives and substitute into the left-hand side:
7. $$\frac{dy}{dx} = 4C e^{4x}, \quad \frac{d^2y}{dx^2} = 16C e^{4x}$$
8. $$16C e^{4x} - 3(4C e^{4x}) + 2(C e^{4x}) = 6C e^{4x}$$
9. 4. Equate to right-hand side to solve for C:
10. $$6C e^{4x} = 6 e^{4x} \implies C = 1$$
11. 5. The general solution is $y = CF + PI$:
12. $$y = A e^x + B e^{2x} + e^{4x}$$

## Common pitfalls

- **Wrong:** Calculating the integrating factor without rearranging first-order DE to standard form.
  - Why it fails: If the coefficient of $\frac{dy}{dx}$ is not 1, $P(x)$ will be wrong leading to an incorrect solution.
  - Correct: Always divide through by the coefficient of $\frac{dy}{dx}$ to get it equal to 1 before calculating $I(x)$.
- **Wrong:** Writing $y = A e^{mx} + B e^{mx} = (A+B)e^{mx}$ for repeated root auxiliary equation.
  - Why it fails: This reduces to only one arbitrary constant, which is incorrect for a second-order equation.
  - Correct: For repeated root $m$, use $y = (A + Bx)e^{mx}$ which has two independent arbitrary constants.
- **Wrong:** Using the same PI guess even when $f(x)$ matches the complementary function.
  - Why it fails: Substituting a PI that is a solution of the homogeneous equation will give 0 on the left-hand side, which can never equal $f(x) \neq 0$.
  - Correct: Multiply your PI guess by $x$ (or $x^2$ for a repeated root) to get a linearly independent solution, then substitute.
- **Wrong:** Forgetting to add the constant of integration after integrating first-order DE.
  - Why it fails: Leaving out $C$ means you lose the arbitrary constant required for a general solution, and cannot find the correct particular solution for boundary conditions.
  - Correct: Add $C$ immediately after integrating, before applying any boundary conditions to solve for its value.

## Cheatsheet

| Equation Type | General Solution |
| --- | --- |
| First-order linear: $\frac{dy}{dx} + P(x)y = Q(x)$ | $I = e^{\int P dx}, \quad y = \frac{1}{I}\left(\int I Q dx + C\right)$ |
| Homogeneous 2nd order: distinct real roots $m_1 \neq m_2$ | $y = A e^{m_1 x} + B e^{m_2 x}$ |
| Homogeneous 2nd order: repeated root $m$ | $y = (A + Bx) e^{m x}$ |
| Homogeneous 2nd order: complex roots $\alpha \pm i\beta$ | $y = e^{\alpha x} (A \cos \beta x + B \sin \beta x)$ |
| Non-homogeneous 2nd order | $y = \text{Complementary Function} + \text{Particular Integral}$ |

## What's next

Differential equations are one of the most heavily examined topics in CIE 9231, appearing in both Further Pure papers and often combined with mechanics modelling questions. Solid mastery of the methods covered here is critical to securing high marks in the exam. This topic forms the foundation for more advanced differential equation topics in further study, and for applied problems across A-Level Further Mathematics. Next, you can explore how to use these methods to model real-world systems, or move on to other core FP2 topics.

- [Further Mechanics](https://www.owlsprep.com/study/cie-9231-u3-overview/)
- [Momentum and Impulse](https://www.owlsprep.com/study/cie-9231-u3-momentum-and-impulse/)

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