📑Example 1: Separable Equations For a separable differential equation, the variables can be separated on either side of the equation:

$\frac{dy}{dx} = g(x) h(y)$

The general solution involves integrating both sides:

$\int \frac{1}{h(y)} \, dy = \int g(x) \, dx$

📑Example: Solve the differential equation:

$\frac{dy}{dx} = xy$

Solution:

1. Separate the variables:

$\frac{1}{y} \, dy = x \, dx$

2. Integrate both sides:

$\int \frac{1}{y} \, dy = \int x \, dx$

$\ln |y| = \frac{x^2}{2} + C$

Where $C$ is the constant of integration.

3. Solve for $y$ :

$y = e^{\frac{x^2}{2} + C} = e^C \cdot e^{\frac{x^2}{2}}$

Since $e^C$ is just another constant, let's call it $C_1$ :

$y = C_1 e^{\frac{x^2}{2}}$

This is the general solution.

📑Example 2: Homogeneous Second-Order Differential Equations Consider:

$\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 0$

Solution:

4. Find the characteristic equation:

$m^2 - 3m + 2 = 0$

5. Solve the quadratic equation:

$(m - 1)(m - 2) = 0$

So, $m = 1$ and $m = 2$ .

6. Write the general solution: For distinct roots $m_1$ and $m_2$ , the general solution is:

$y(x) = C_1 e^{m_1 x} + C_2 e^{m_2 x}$

Substituting $m_1 = 1$ and $m_2 = 2$ :

$y(x) = C_1 e^{x} + C_2 e^{2x}$

This is the general solution.

The general solution of a differential equation represents the family of all possible solutions, typically including one or more arbitrary constants. For first-order equations, the general solution is found by integration, while for second-order equations with constant coefficients, it involves solving a characteristic equation. Understanding how to derive these solutions is essential for solving problems that model real-world phenomena in physics, engineering, and other fields.