A scalar first-order differential equation typically takes the form:

$\frac{dy}{dx} = f(x, y)$

General Steps to Solve a Scalar First-Order Differential Equation

1. Identify the Type of Differential Equation: The first step is to recognize the specific form of the equation. Common forms include:

   • Separable Differential Equations: $\frac{dy}{dx} = g(x)h(y)$
   • Linear Differential Equations: $\frac{dy}{dx} + P(x)y = Q(x)$
   • Exact Differential Equations: $M(x, y)dx + N(x, y)dy = 0$

2. Solve Based on the Type:

   • Separable Equations: Separate the variables and integrate: $\frac{1}{h(y)} dy = g(x) dx$ Integrate both sides to find the solution.

   • Linear Equations: Use an integrating factor: $I(x) = e^{\int P(x)dx}$ Multiply the entire equation by the integrating factor, and then solve by integrating.

   • Exact Equations: If the equation is exact, solve using potential functions. Check if $M_y = N_x$ for exactness, then find the potential function $\psi(x, y)$.

3. Apply Initial Conditions (if any): If the problem provides an initial condition $y(x_0) = y_0$, use this to solve for the constant of integration.

4. Express the Solution Explicitly: Where possible, rearrange the equation to solve for $y(x)$ explicitly. In some cases, implicit solutions may be acceptable.

Example: Solve the separable equation

$\frac{dy}{dx} = 3x^2y$

Step 1: Separate variables: $\frac{1}{y}dy = 3x^2 dx$

Step 2: Integrate both sides: $\int \frac{1}{y}dy = \int 3x^2 dx$ $\ln|y| = x^3 + C$

Step 3: Solve for $y$: $y = e^{x^3 + C} = e^C e^{x^3}$ Let $C_1 = e^C$, so the solution becomes: $y = C_1 e^{x^3}$

If an initial condition is provided (e.g., $y(0) = 2$), you can find the constant $C_1$.

Do you need more details or have any questions? Here are some related questions:

1. How do you solve exact differential equations?
2. What is the method for solving homogeneous first-order differential equations?
3. Can you solve non-linear first-order differential equations?
4. What are the applications of first-order differential equations in physics?
5. How do integrating factors help in solving linear first-order differential equations?

Tip: Always check for the possibility of separability first when dealing with scalar first-order differential equations—it simplifies the process considerably.