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Ordinary Differential Equation

Ordinary Differential Equation

A differential equation which does not include any partial derivatives.

 

 

See also

Partial differential equation

Key Formula

an(x)dnydxn+an−1(x)dn−1ydxn−1+⋯+a1(x)dydx+a0(x) y=g(x)a_n(x)\frac{d^n y}{dx^n} + a_{n-1}(x)\frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1(x)\frac{dy}{dx} + a_0(x)\,y = g(x)
Where:
  • xx = The single independent variable
  • yy = The unknown function of x
  • dkydxk\frac{d^k y}{dx^k} = The k-th ordinary derivative of y with respect to x
  • a0,a1,…,ana_0, a_1, \ldots, a_n = Coefficient functions that may depend on x
  • g(x)g(x) = A known function (the forcing term); if g(x) = 0, the ODE is called homogeneous
  • nn = The order of the ODE, determined by the highest derivative present

Worked Example

Problem: Verify that y = 3e^(2x) is a solution to the ordinary differential equation dy/dx − 2y = 0.
Step 1: Identify the ODE and note its order. The highest derivative is dy/dx, so this is a first-order ODE.
dydx−2y=0\frac{dy}{dx} - 2y = 0
Step 2: Compute the derivative of the proposed solution y = 3e^(2x).
dydx=3⋅2 e2x=6e2x\frac{dy}{dx} = 3 \cdot 2\,e^{2x} = 6e^{2x}
Step 3: Substitute y and dy/dx into the left side of the equation.
6e2x−2(3e2x)=6e2x−6e2x=06e^{2x} - 2(3e^{2x}) = 6e^{2x} - 6e^{2x} = 0
Step 4: The left side equals the right side (0), so the equation is satisfied for all x.
0=0✓0 = 0 \quad \checkmark
Answer: Since substituting y = 3e^(2x) makes both sides equal, it is indeed a solution to the ODE dy/dx − 2y = 0.

Another Example

Problem: Solve the second-order ODE d²y/dx² + y = 0, which models simple harmonic motion.
Step 1: This is a second-order linear homogeneous ODE with constant coefficients. Write the characteristic equation by replacing d²y/dx² with r² and y with 1.
r2+1=0r^2 + 1 = 0
Step 2: Solve for r.
r=± ir = \pm\, i
Step 3: Complex roots r = ±i give the general solution using sine and cosine, where C₁ and C₂ are arbitrary constants.
y=C1cos⁡x+C2sin⁡xy = C_1 \cos x + C_2 \sin x
Answer: The general solution is y = C₁ cos x + C₂ sin x, a combination of oscillating functions familiar from physics (springs, pendulums, circuits).

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 ordinary derivatives. A partial differential equation (PDE) involves a function of two or more independent variables and its partial derivatives. For example, dy/dx = 3y is an ODE (one variable x), while ∂u/∂t = k ∂²u/∂x² (the heat equation) is a PDE (two variables t and x).
What does the 'order' of an ODE mean?
The order of an ODE is the highest derivative that appears in the equation. For instance, dy/dx + y = 0 is first-order because the highest derivative is the first derivative, while d²y/dx² + y = 0 is second-order because the highest derivative is the second derivative. The order strongly influences what solution methods apply.

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

An ODE has one independent variable, so all derivatives are ordinary (d/dx). A PDE has two or more independent variables, so it involves partial derivatives (∂/∂x, ∂/∂t, etc.). ODEs typically yield solutions that are functions of one variable, while PDEs yield solutions that are functions of multiple variables. ODEs are generally easier to solve and are studied first in most curricula.

Why It Matters

ODEs model countless real-world processes where change depends on a single variable, such as population growth over time, radioactive decay, the motion of a spring, or current in an electrical circuit. Newton's second law (F = ma) is itself an ODE relating force to acceleration (the second derivative of position). Mastering ODEs is essential groundwork for engineering, physics, biology, and economics.

Common Mistakes

Mistake: Confusing the order of an ODE with its degree. Students sometimes call d²y/dx² + y = 0 a 'degree 2' equation.
Correction: The order is the highest derivative present (here, 2nd). The degree is the power to which that highest derivative is raised (here, 1). So this equation is second-order, first-degree.
Mistake: Forgetting the arbitrary constant(s) when writing the general solution.
Correction: A first-order ODE has one arbitrary constant, a second-order ODE has two, and in general an nth-order ODE has n arbitrary constants. Omitting them gives only a particular solution, not the general solution.

Related Terms