Linear

Definition

A first-order linear differential equation is an equation of the form $\frac{dy}{dx} + P(x)y = Q(x)$. It is called linear because both the dependent variable and its derivative appear to the first power and are not multiplied together.

5 Must Know Facts For Your Next Test

1. The standard form for a first-order linear differential equation is $\frac{dy}{dx} + P(x)y = Q(x)$.
2. The integrating factor method is commonly used to solve first-order linear differential equations.
3. The general solution of a first-order linear differential equation can be written as $y = \frac{1}{\mu(x)} \left( \int \mu(x)Q(x)dx + C \right)$, where $\mu(x) = e^{\int P(x)dx}$.
4. If $P(x)$ or $Q(x)$ are zero, the differential equation simplifies significantly.
5. First-order linear equations model various real-world phenomena, including radioactive decay and cooling processes.

Review Questions

• What is the standard form of a first-order linear differential equation?
• Explain the purpose of an integrating factor in solving a first-order linear differential equation.
• Provide an example of a real-world problem that can be modeled by a first-order linear differential equation.

Related terms

Integrating Factor:

A function, typically denoted as $\mu(x)$, used to simplify solving first-order linear differential equations by making them exact.

Homogeneous Equation: A differential equation where $Q(x) = 0$, resulting in the form $\frac{dy}{dx} + P(x)y = 0$.

Particular Solution:

A solution to a non-homogeneous differential equation that includes no arbitrary constants and satisfies the specific conditions given in the problem.