Solving First-Order Separable Differential Equations

Concept Overview

This question tests the fundamental technique for solving first-order ordinary differential equations where the variables can be separated. The core idea is to rearrange the equation so that all terms involving one variable (and its differential) are on one side, and all terms involving the other variable (and its differential) are on the other. Once separated, we integrate both sides to find the general solution, remembering to include the constant of integration.

Step 1: Identify if the differential equation is separable. A first-order differential equation of the form $\frac{dy}{dx} = f(x, y)$ is separable if it can be written as $\frac{dy}{dx} = g(x)h(y)$. This means the function on the right-hand side can be expressed as a product of a function of $x$ only and a function of $y$ only.

Step 2: Separate the variables. If the equation is separable, we rearrange it to group all $y$ terms with $dy$ and all $x$ terms with $dx$. This is done by dividing by $h(y)$ (assuming $h(y) \neq 0$) and multiplying by $dx$. $\frac{dy}{h(y)} = g(x) dx$ This step isolates the variables, preparing the equation for integration.

Step 3: Integrate both sides of the separated equation. Once the variables are separated, we integrate both sides with respect to their respective variables. $\int \frac{1}{h(y)} dy = \int g(x) dx$ This integration process will yield expressions involving $y$ and $x$.

Step 4: Introduce the constant of integration. When performing indefinite integration, we must add a constant of integration, usually denoted by $C$. It is sufficient to add the constant to only one side of the equation, typically the side involving the independent variable ($x$ in this case). $\int \frac{1}{h(y)} dy = \int g(x) dx + C$ This constant $C$ represents the family of solutions to the differential equation.

Step 5: Solve for $y$ in terms of $x$ (if possible) to obtain the general solution. After integration, the resulting equation implicitly or explicitly relates $y$ and $x$. If possible, we solve this equation for $y$ to express the general solution explicitly as $y = F(x, C)$. If solving for $y$ is difficult or not required, the implicit form is also considered the general solution.

For example, consider the differential equation $\frac{dy}{dx} = \frac{x}{y}$. Here, $g(x) = x$ and $h(y) = \frac{1}{y}$. Separating variables: $y \, dy = x \, dx$. Integrating both sides: $\int y \, dy = \int x \, dx + C$. Performing the integration: $\frac{y^2}{2} = \frac{x^2}{2} + C$. This is the implicit general solution. We can rewrite it as $y^2 - x^2 = 2C$. Let $K = 2C$ (another arbitrary constant), so $y^2 - x^2 = K$.

Key Takeaways:

• A differential equation is separable if it can be written as $\frac{dy}{dx} = g(x)h(y)$.
• The method involves isolating $y$ terms with $dy$ and $x$ terms with $dx$ on opposite sides of the equation.
• Integrate both sides of the separated equation and add a single constant of integration ($C$) to one side.
• The resulting equation, after integration and simplification, is the general solution.

Answer: The general solution is obtained by separating variables, integrating both sides, and adding a constant of integration.