The complete definition of linear first-order differential equations

📚 Definition of Linear First-Order Differential Equations

A linear first-order differential equation is an equation that can be written in the form:

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

where $y$ is a function of $x$, and $P(x)$ and $Q(x)$ are functions of $x$ only. The key characteristics are that the dependent variable $y$ and its derivative appear only to the first power, and there are no products of $y$ with itself or its derivatives.

📜 History and Background

Differential equations, including linear first-order equations, have been studied since the invention of calculus in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. These equations became crucial in describing physical phenomena. The study of linear equations advanced rapidly in the 18th and 19th centuries with mathematicians like Euler, Bernoulli, and Lagrange contributing significantly to their theory and methods of solution.

🔑 Key Principles

• 🔍 Standard Form: Ensure the equation is in the standard form $\frac{dy}{dx} + P(x)y = Q(x)$ before proceeding.
• 📈 Integrating Factor: Calculate the integrating factor $\mu(x) = e^{\int P(x) dx}$. This factor helps to simplify the equation.
• ➗ Multiply: Multiply both sides of the equation by the integrating factor $\mu(x)$.
• 🧮 Integration: Integrate both sides of the equation with respect to $x$. The left side will simplify to the derivative of the product of $y$ and the integrating factor.
• ✅ Solve for y: Isolate $y$ to find the general solution of the differential equation.

⚙️ Method of Solution

To solve a linear first-order differential equation:

1. Step 1: Write the equation in the standard form: $\frac{dy}{dx} + P(x)y = Q(x)$.
2. Step 2: Find the integrating factor: $\mu(x) = e^{\int P(x) dx}$.
3. Step 3: Multiply both sides of the equation by $\mu(x)$: $\mu(x) \frac{dy}{dx} + \mu(x) P(x)y = \mu(x) Q(x)$.
4. Step 4: Observe that the left side is the derivative of $(\mu(x)y)$ with respect to $x$: $\frac{d}{dx}(\mu(x)y) = \mu(x) Q(x)$.
5. Step 5: Integrate both sides with respect to $x$: $\int \frac{d}{dx}(\mu(x)y) dx = \int \mu(x) Q(x) dx$, which simplifies to $\mu(x)y = \int \mu(x) Q(x) dx$.
6. Step 6: Solve for $y$: $y = \frac{1}{\mu(x)} \int \mu(x) Q(x) dx$.

🧪 Real-world Examples

• 🌡️ Cooling/Heating: Newton's Law of Cooling, $\frac{dT}{dt} = k(T - T_s)$, where $T$ is the temperature of an object, $T_s$ is the surrounding temperature, and $k$ is a constant.
• 🧮 Radioactive Decay: $\frac{dN}{dt} = -\lambda N$, where $N$ is the number of radioactive atoms, and $\lambda$ is the decay constant.
• 💧 Mixing Problems: Analyzing the amount of salt in a tank as brine is added and removed.
• ⚡ Electrical Circuits: Analyzing current in an RC or RL circuit.

💡 Conclusion

Linear first-order differential equations are a fundamental tool in mathematical modeling, offering insights into numerous physical systems. Understanding their properties and solution methods is essential for students and professionals in science and engineering. Mastering these equations provides a solid foundation for tackling more complex mathematical problems.