Linear Differential Equation
A linear differential equation is an ODE in which the unknown function and all its derivatives appear linearly (to the first power only, with no products between them), with coefficients that may be functions of the independent variable. The standard first-order linear form is dy/dx + P(x)y = Q(x), solved using an integrating factor. Linear differential equations obey the superposition principle, making them tractable analytically and foundational in engineering, electrical circuit analysis, and quantum mechanics.
Key Formula
dy/dx + P(x)·y = Q(x), integrating factor μ(x) = e^(∫P(x)dx)
LaTeX: \frac{dy}{dx} + P(x)\,y = Q(x), \quad \mu(x) = e^{\int P(x)\,dx}
| Symbol | Meaning | Unit |
|---|---|---|
| P(x) | Coefficient function of the dependent variable term | dimensionless |
| Q(x) | Non-homogeneous (forcing) function | dimensionless |
| μ(x) | Integrating factor | dimensionless |
Worked Example
Problem
Solve the linear ODE: dy/dx + (2/x)y = 4x², with initial condition y(1) = 1.
Solution
Step 1: Identify P(x) = 2/x, Q(x) = 4x². Step 2: Compute integrating factor: μ(x) = e^(∫2/x dx) = e^(2 ln x) = x². Step 3: Multiply through by μ: x²(dy/dx) + 2xy = 4x⁴ → d/dx(x²y) = 4x⁴. Step 4: Integrate both sides: x²y = (4x⁵)/5 + C. Step 5: Apply IC y(1) = 1: 1 = 4/5 + C → C = 1/5. Step 6: Final solution: y = (4x³)/5 + 1/(5x²).
Answer
y = (4x³)/5 + 1/(5x²)
Steps to Solve a First-Order Linear ODE
| Step | Action | Formula / Result |
|---|---|---|
| 1 | Write in standard form | dy/dx + P(x)y = Q(x) |
| 2 | Compute integrating factor | μ(x) = e^(∫P(x)dx) |
| 3 | Multiply both sides by μ | d/dx[μ(x)y] = μ(x)Q(x) |
| 4 | Integrate both sides | μ(x)y = ∫μ(x)Q(x)dx + C |
| 5 | Solve for y | y = [∫μ(x)Q(x)dx + C] / μ(x) |
| 6 | Apply initial condition | Determine value of C |
Interactive Tools
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Related Terms
Differential Equation
A differential equation is an equation that relates a function to one or more of its derivatives, describing how a quantity changes with respect to one or more independent variables. Ordinary differential equations (ODEs) involve functions of a single variable, while partial differential equations (PDEs) involve functions of multiple variables. Differential equations are fundamental to modeling physical, biological, and engineering systems — from Newton's laws of motion to population dynamics and heat conduction.
Separable Differential Equation
A separable differential equation is a first-order ODE in which the variables can be algebraically rearranged so that all terms involving the dependent variable appear on one side and all terms involving the independent variable appear on the other side, allowing direct integration of each side independently. The standard form is dy/dx = g(x)·h(y), which rearranges to (1/h(y))dy = g(x)dx. Separable equations model exponential growth, radioactive decay, Newton's law of cooling, and many other phenomena.
Euler's Method (ODE)
Euler's method is a first-order numerical procedure for approximating solutions to ordinary differential equations given an initial value. Starting from a known point, the method steps forward along the tangent line to the solution curve in increments of step size h, using the ODE's derivative expression to determine the direction. Although simple to implement, Euler's method accumulates error with larger step sizes and has been superseded by more accurate methods (such as Runge–Kutta), but it remains the foundational teaching example for numerical ODE solvers.
From Latin "linearis" (of or belonging to a line). The term "linear" reflects the property that solutions satisfy superposition — if y₁ and y₂ are solutions of the homogeneous equation, so is any linear combination c₁y₁ + c₂y₂. The integrating factor method was systematized by Gottfried Leibniz and Alexis Clairaut in the 18th century.