Differential Equations
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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