Differential Equation

Differential Equation

An equation showing a relationship between a function and its derivative(s). For example, dy/dx + y = 0 is a differential equation with solutions y = Ce^–x.

Key Formula

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

Where:

$y$ = The unknown function of x that you are solving for
$x$ = The independent variable
$\frac{dy}{dx}$ = The first derivative of y with respect to x
$f(x, y)$ = A given function that defines the relationship between x, y, and the derivative

Worked Example

Problem: Solve the first-order differential equation dy/dx = 3x², given that y(0) = 5.

Step 1: Recognize that this equation can be solved by direct integration, since the right side depends only on x.

\frac{dy}{dx} = 3x^2

Step 2: Integrate both sides with respect to x.

y = \int 3x^2\, dx = x^3 + C

Step 3: Apply the initial condition y(0) = 5 to find the constant C.

5 = (0)^3 + C \implies C = 5

Step 4: Write the particular solution.

y = x^3 + 5

Answer: The solution is y = x³ + 5.

Another Example

This example involves separating variables when the derivative depends on y, unlike the first example where the right side depended only on x and could be solved by direct integration.

Problem: Solve the separable differential equation dy/dx = 2y, given that y(0) = 3.

Step 1: Separate variables by moving all y terms to one side and all x terms to the other.

\frac{dy}{y} = 2\, dx

Step 2: Integrate both sides.

\int \frac{1}{y}\, dy = \int 2\, dx \implies \ln|y| = 2x + C_1

Step 3: Exponentiate both sides to solve for y. Combine the constant into a single positive constant A.

|y| = e^{2x + C_1} = e^{C_1} e^{2x} \implies y = Ae^{2x}

Step 4: Apply the initial condition y(0) = 3 to determine A.

3 = Ae^{0} = A \implies A = 3

Step 5: Write the particular solution.

y = 3e^{2x}

Answer: The solution is y = 3e^{2x}.

Frequently Asked Questions

What is the difference between an ordinary differential equation and a partial differential equation?

An ordinary differential equation (ODE) involves a function of one independent variable and its derivatives. A partial differential equation (PDE) involves a function of two or more independent variables and its partial derivatives. For instance, dy/dx = 2y is an ODE, while ∂u/∂t = k(∂²u/∂x²) is a PDE.

What does it mean to solve a differential equation?

Solving a differential equation means finding a function (or family of functions) that satisfies the equation when substituted in. The general solution contains arbitrary constants representing a whole family of curves. A particular solution is found by applying initial or boundary conditions to determine those constants.

What is the order of a differential equation?

The order of a differential equation is the highest-order derivative that appears in the equation. For example, dy/dx = x has order 1, while d²y/dx² + 3dy/dx = 0 has order 2. Higher-order equations typically require more initial conditions to determine a unique solution.

Ordinary Differential Equation (ODE) vs. Partial Differential Equation (PDE)

	Ordinary Differential Equation (ODE)	Partial Differential Equation (PDE)
Number of independent variables	One (e.g., x or t)	Two or more (e.g., x and t)
Derivative type	Ordinary derivatives (dy/dx)	Partial derivatives (∂u/∂x)
Typical example	dy/dx + y = 0	∂²u/∂x² + ∂²u/∂y² = 0
Solution form	A function of one variable, y(x)	A function of multiple variables, u(x, y)
When encountered	First calculus/DE course	Advanced math and physics courses

Why It Matters

Differential equations appear throughout science, engineering, and economics because they model how quantities change over time or space. In physics, Newton's second law (F = ma) is itself a differential equation relating position to force. Students first encounter them in calculus courses, and they become central in courses on physics, biology (population models), and engineering (circuit analysis, heat transfer).

Common Mistakes

Mistake: Forgetting the constant of integration when finding the general solution.

Correction: Every indefinite integration step introduces an arbitrary constant C. Without it, you only get one specific solution instead of the full family. The constant is later determined by an initial or boundary condition.

Mistake: Confusing the order of a differential equation with its degree.

Correction: The order is the highest derivative present (e.g., d²y/dx² means order 2). The degree is the power of that highest-order derivative after the equation is cleared of fractions and radicals. For example, (d²y/dx²)³ + dy/dx = 0 has order 2 and degree 3.

Related Terms

Equation — General concept that differential equations extend
Function — The unknown quantity solved for in a DE
Derivative — Core operation appearing in every DE
Solution — The function satisfying the differential equation
Ordinary Differential Equation — DE with one independent variable
Partial Differential Equation — DE with multiple independent variables
Order of a Differential Equation — Highest derivative order in the equation
Initial Value Problem — DE paired with conditions at a starting point