Differential Equations — Study Notes

1. Introduction

A differential equation is an equation that involves an unknown function and its derivatives. Differential equations are the language of change in science: they describe population growth, radioactive decay, the motion of planets, the flow of heat, and the dynamics of electrical circuits. Learning to recognise and solve differential equations equips the student with one of the most broadly applicable techniques in all of mathematics.

This chapter begins with the definitions of order and degree of a differential equation and the general and particular solutions. It then studies the main classes of first-order equations: equations in which the variables can be separated, homogeneous differential equations, and linear differential equations. The methods of solution for each type are developed with worked examples, and the chapter also shows how to form a differential equation from a given family of curves by eliminating the arbitrary constants.

The chapter is highly examinable, with questions on order and degree, variable separation, homogeneous equations, and linear equations appearing year after year. Because the techniques are formula-driven, systematic practice converts this chapter into a guaranteed source of marks. The chapter also forms the conceptual bridge between the differential calculus of earlier chapters and the applied modelling problems that appear in higher studies.

2. Order and Degree of a Differential Equation

The order of a differential equation is the order of the highest-order derivative appearing in it. The degree of a differential equation is the power of the highest-order derivative, provided the equation is expressed as a polynomial in the derivatives and their coefficients.

For example, $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 0$ has order 2 and degree 1.

The equation $\left(\frac{d^3y}{dx^3}\right)^2 + \sin x = 0$ has order 3 and degree 2.

The degree is defined only when the derivatives occur with non-negative integer powers; equations involving trigonometric or logarithmic functions of derivatives have no defined degree.

3. General and Particular Solutions

A solution of a differential equation is a relation between x and y, free of derivatives, that satisfies the equation identically. The general solution contains arbitrary constants, and each choice of these constants gives a particular solution.

For example, the differential equation $\frac{dy}{dx} = 2x$ has general solution $y = x^2 + C$. If the additional condition $y(0) = 3$ is given, then $C = 3$ and the particular solution is $y = x^2 + 3$. The conditions used to fix the constants are called initial or boundary conditions.

4. Formation of Differential Equations

To form a differential equation from a family of curves with n arbitrary constants: 1. Differentiate the given equation of the family n times to eliminate the constants. 2. The result is a differential equation of order n (or less) that the family satisfies.

Example

The family of straight lines $y = mx$ has one constant m. Differentiating, $\frac{dy}{dx} = m$, and eliminating m:

$$y = x\frac{dy}{dx} \quad \text{or} \quad \frac{dy}{dx} = \frac{y}{x}$$

5. Methods of Solving First-Order Differential Equations

Method 1: Variables Separable

A differential equation of the form

$$\frac{dy}{dx} = f(x)g(y)$$

is solved by separating the variables and integrating:

$$\int \frac{dy}{g(y)} = \int f(x)\, dx + C$$

Example

$$\frac{dy}{dx} = \frac{x}{y} \implies \int y\, dy = \int x\, dx \implies \frac{y^2}{2} = \frac{x^2}{2} + C$$

So $y^2 - x^2 = 2C$, a family of rectangular hyperbolas.

Method 2: Homogeneous Differential Equations

A differential equation $\frac{dy}{dx} = f(x, y)$ is homogeneous if f(x, y) can be expressed as a function of $\frac{y}{x}$, i.e., f(x, y) = g(y/x). To solve:

1. Put $y = vx$, so that $\frac{dy}{dx} = v + x\frac{dv}{dx}$.
2. Substitute into the equation, which then separates into functions of v and x.
3. Integrate and substitute back $v = y/x$.

Example

$$\frac{dy}{dx} = \frac{x + y}{x - y}$$

Put $y = vx$. Then

$$v + x\frac{dv}{dx} = \frac{1 + v}{1 - v}$$

This separates to give a solution involving $\tan^{-1}(y/x)$ and a logarithm.

Method 3: Linear Differential Equations

A first-order linear differential equation has the form

$$\frac{dy}{dx} + P(x)y = Q(x)$$

It is solved using the integrating factor (IF):

$$\text{IF} = e^{\int P(x)\, dx}$$

Multiplying through by the IF makes the left side the derivative of $y \cdot \text{IF}$, so

$$y \cdot \text{IF} = \int Q(x) \cdot \text{IF}\, dx + C$$

Example

$$\frac{dy}{dx} + 2y = e^{-x}$$

Here P = 2, so IF = $e^{2x}$. Then

$$y e^{2x} = \int e^{-x} e^{2x} dx = \int e^x dx = e^x + C$$

$$y = e^{-x} + Ce^{-2x}$$

Quick Revision Tables

Table 1: Types and Methods

Table 2: Order, Degree and Solution

Table 3: Common Solution Forms

Mind Map

Important Diagrams (SVG)

Diagram 1: Classification Flowchart of First-Order Equations

Diagram 2: Slope Field of dy/dx = 2x with Family of Solutions

Common Mistakes

1. Declaring the degree of an equation containing sin(dy/dx) or e^(dy/dx); the degree is undefined unless the equation is polynomial in the derivatives.
2. Confusing the order of an equation with the degree. Order is the order of the highest derivative; degree is its power.
3. Forgetting to substitute back y/x for v after solving a homogeneous equation, leaving the answer in terms of v.
4. Missing the arbitrary constant C in the general solution of a separable equation; the answer is incomplete without it.
5. In linear equations, forgetting to divide the entire equation by the coefficient of dy/dx first so that the equation has the standard form dy/dx + Py = Q.
6. Forgetting the integrating factor is e^(int P dx), not e^(int Q dx); the exponent uses the coefficient of y.
7. In the linear equation formula, writing y(IF) = int Q(IF)dx but then failing to divide by the IF at the end.
8. Eliminating constants incorrectly when forming a differential equation; all constants must be eliminated, not some.
9. Substituting y = vx but failing to differentiate y as v + x(dv/dx) correctly in homogeneous equations.
10. Applying the separation of variables method to an equation that is not separable; check that the variables genuinely separate.

Exam Tips

1. For order and degree questions, first write the equation in polynomial form if necessary, then read off the order and the power of the highest derivative.
2. To form a differential equation, count the arbitrary constants; this tells you how many times to differentiate before eliminating them.
3. In separable equations, integrate each side with respect to its own variable and combine constants into a single C.
4. For homogeneous equations, test homogeneity by replacing x with tx and y with ty; the equation must be unchanged for homogeneity to hold.
5. In linear equations, always compute the integrating factor first, multiply through, and verify that the left side collapses to d/dx(y * IF).
6. Apply initial conditions at the very end by substituting the given values into the general solution to fix the constant.
7. Verify the final answer by differentiating it and substituting back into the original differential equation; this catches algebraic slips.
8. Practise recognising dy/dx = (a x + b y + c)/(p x + q y + r) type problems where a shift of origin makes the equation homogeneous.

Conclusion

Differential equations express the fundamental principle that the rate of change of a quantity is governed by its current state, and solving such equations recovers the quantity itself. The classification into order and degree gives a first handle on any equation, and the three principal solution methods, variable separation, the homogeneous substitution, and the integrating factor for linear equations, cover the full first-order syllabus. Forming differential equations from families of curves shows the converse construction and completes the picture. Because the methods are mechanical once the type is identified, this is a chapter where careful, structured practice reliably produces full marks. The ideas also seed the modelling mindset essential for physics, biology, economics, and engineering, making differential equations one of the most valuable chapters in the course.