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Differential Equations

Differential Equations

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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 = e2x is a solution to dy/dx = 2y by finding its first and second derivatives and plugging them into the original equation.

  • dy/dx = 2e2x
  • d2y/dx2 = 4e2x

After substitution, we get 2e2x = 2e2x, 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

y2 = x2 + C

y = √(x2 + C)

The same solution can be obtained by using the substitution u = x/y; in this case, we get x2 + 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.

y2 = x2 + C

4 = 1 + C

C = 3

Therefore, the specific solution is y = √(x2 + 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

-y2 = x2 + C

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

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