## Differential Equations: A Comprehensive Guide

Differential equations are a type of mathematical equations that involve the derivatives of a function. Although they may seem daunting at first, having the ability to solve them can be incredibly useful in various real-life applications. In this article, we will cover the basics of differential equations, including how to verify a solution, solve first-order separable equations, sketch solution curves, and model real-life scenarios.

## Understanding Differential Equations

Differential equations consist of multiple derivatives of a function. The order of a differential equation is determined by the highest order of any derivative present. These equations can represent a wide range of systems that involve changes with respect to time, position, and other variables.

## Verifying Solutions of Differential Equations

When given a differential equation, we can verify a potential solution by finding all its derivatives and substituting them into the equation. For instance, we can verify that **y = e**^{2x} is a solution to **dy/dx = 2y** by finding its first and second derivatives and plugging them into the original equation.

**dy/dx = 2e**^{2x}**d**^{2}y/dx^{2} = 4e^{2x}

After substitution, we get **2e**^{2x} = 2e^{2x}, which verifies that the solution is correct.

## Solving Differential Equations

At the A level, we only need to know how to solve first-order separable ordinary differential equations. These equations have the form **dy/dx = f(x)g(y)** and can be solved by separating the variables, integrating, and then solving for **y**.

For example, let's find the general solution to **dy/dx = 2x/y**.

**dy/dx = 2x/y**

**ydy = 2xdx**

**∫ydy = ∫2xdx**

**y**^{2} = x^{2} + C

**y = √(x**^{2} + C)

The same solution can be obtained by using the substitution **u = x/y**; in this case, we get **x**^{2} + C = ln|y|. Regardless of the method used, the solution will always have an unknown constant.

For instance, let's find the solution to **dy/dx = 2x/y** with the boundary condition **y(1) = 2**.

**y**^{2} = x^{2} + C

**4 = 1 + C**

**C = 3**

Therefore, the specific solution is **y = √(x**^{2} + 3).

## Sketching Solution Curves for Differential Equations

Another important skill when working with differential equations is being able to sketch solution curves. For example, let's find the general solution to **dy/dx = -x/y** and sketch four different particular solutions.

**-ydy = xdx**

**-y**^{2} = x^{2} + C

The graph below shows the solution curves when **C = -1, 0, 1, 2**.

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