First-Order Differential Equations

TL;DR

First-order differential equations relate a function to its first derivative, describing how quantities change. You'll learn to solve them by separating variables or using an integrating factor for linear cases. These equations are fundamental for modeling real-world processes like population growth or cooling.

1. The Mental Model

Think of a first-order differential equation as a rule that tells you how fast something is changing, based on its current state. Your goal is to find the actual "state" function that follows this rule.

2. The Core Material

First-order differential equations are equations involving an unknown function $y(x)$ and its first derivative $dy/dx$. They typically look like $dy/dx = f(x, y)$.

Separable Equations

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A differential equation is separable if you can rearrange it so all terms involving $y$ (and $dy$) are on one side, and all terms involving $x$ (and $dx$) are on the other.

General form: $dy/dx = g(x)h(y)$

How to solve:
1. Rewrite as $dy/h(y) = g(x)dx$.
2. Integrate both sides: $\int (1/h(y))dy = \int g(x)dx$.
3. Solve for $y$ if possible. Don't forget the constant of integration, $C$.

Linear Equations

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A first-order differential equation is linear if it can be written in the form: $dy/dx + P(x)y = Q(x)$. Here, $P(x)$ and $Q(x)$ are functions of $x$ (or constants).

How to solve (using an Integrating Factor):
1. Identify $P(x)$ and $Q(x)$.
2. Calculate the integrating factor, $I(x) = e^{\int P(x)dx}$.
3. Multiply the entire linear equation by $I(x)$: $I(x)(dy/dx) + I(x)P(x)y = I(x)Q(x)$.
4. The left side is now the derivative of a product: $d/dx [I(x)y]$.
5. So, $d/dx [I(x)y] = I(x)Q(x)$.
6. Integrate both sides with respect to $x$: $I(x)y = \int I(x)Q(x)dx$.
7. Solve for $y$: $y = (1/I(x))\int I(x)Q(x)dx$. Again, remember $C$.

Here's a flowchart to help you decide which method to use:

graph TD
    A["Start with dy/dx = f(x, y)"] --> B{"Can you separate x and y?"}
    B -- Yes --> C["Rewrite as g(y)dy = h(x)dx"]
    C --> D["Integrate both sides"]
    D --> E["Solve for y"]
    B -- No --> F{"Is it in the form dy/dx + P(x)y = Q(x)?"}
    F -- Yes --> G["Calculate Integrating Factor I(x) = e^(∫P(x)dx)"]
    G --> H["Multiply equation by I(x)"]
    H --> I["Integrate I(x)y = ∫I(x)Q(x)dx"]
    I --> E
    F -- No --> J["Requires advanced methods (beyond this scope)"]

3. Worked Example

Let's solve the differential equation: $dy/dx + (2/x)y = x^2$ for $x > 0$.

1. Identify the type: This is a linear first-order differential equation of the form $dy/dx + P(x)y = Q(x)$, where $P(x) = 2/x$ and $Q(x) = x^2$.

2. Calculate the integrating factor:
   $\int P(x)dx = \int (2/x)dx = 2\ln|x| = \ln(x^2)$ (since $x>0$)
   $I(x) = e^{\int P(x)dx} = e^{\ln(x^2)} = x^2$.

3. Multiply the equation by $I(x)$:
   $x^2(dy/dx) + x^2(2/x)y = x^2(x^2)$
   $x^2(dy/dx) + 2xy = x^4$

4. Recognize the product rule: The left side is $d/dx(x^2y)$.
   So, $d/dx(x^2y) = x^4$.

5. Integrate both sides:
   $\int d/dx(x^2y)dx = \int x^4 dx$
   $x^2y = (x^5/5) + C$

6. Solve for $y$:
   $y = (1/x^2)((x^5/5) + C)$
   $y = (x^3/5) + C/x^2$

This is the general solution to the differential equation.

4. Key Takeaways

• First-order differential equations describe rates of change based on current conditions.
• Separable equations can be rearranged to isolate variables on each side before integrating.
• Linear equations have the form $dy/dx + P(x)y = Q(x)$ and are solved using an integrating factor.
• The integrating factor $I(x) = e^{\int P(x)dx}$ simplifies linear equations for integration.
• Always include the constant of integration, $C$, after performing indefinite integration.
• Check your solution by differentiating it and plugging it back into the original equation.

Common Mistakes to Avoid:
- Forgetting the constant of integration, $C$, which can lead to an incomplete general solution.
- Incorrectly separating variables; make sure $dx$ and $dy$ are on the correct sides with their respective functions.
- Errors in calculating the integral for the integrating factor or the final integration step.
- Not multiplying every term in the linear equation by the integrating factor.

5. Now Try It

Solve the following first-order differential equation: $dy/dx = (x^2 + 1)/y$.

What to do:
1. Determine if it's separable or linear.
2. Apply the appropriate method to find the general solution.
3. Clearly show each step of your integration.

What success looks like:
You should arrive at an equation relating $y^2$ to $x^3 + x + C$.