Calculus How To

Differential Equations

Differential equations are equations that have a derivative. For example, dy/dx = 9x. In elementary algebra, you usually find a single number as a solution, like x = 12. But with differential equations, the solutions are functions. The equations represent the relationship between a varying quantity and it’s rate of change.

Problems with differential equations are asking you to find an unknown function or functions, rather than a number or set of numbers as you would normally find with an equation like f(x) = x2 + 9. For example, the differential equation
dy⁄dx = 10x
is asking you to find the derivative of some unknown function y that is equal to 10x.

Problems, Solutions and Examples

General solutions are where the solution is a function or set of functions. For example, the differential equation dy⁄dx = 2x means that you have to find the derivative of some unknown function y that is equal to 10x. You’ll always need to add a constant (+ C) as these are general (as opposed to particular) solutions. See: How to Find the General Solution for a Differential Equation.

initial value problems are where you are given an initial condition (for example, the initial condition y(0) = 3); You must find a particular solution that satisfies the initial condition. For example, the differential equation
dy⁄dx 19 x2 + 10; y(10) = 5
requires you to find a particular solution (a single function) that satisfies y(10) = 5. See: How to Solve a Differential Equation with an Initial Condition

A particular solution requires you to find a solution that meets the specific constraints given in the question. For example, while
dy⁄dv x3 + 8
requires a general solution (with + C), the differential equation
dy⁄dv x3 + 8; f(0) = 2
requires a particular solution, one that fits the constraint f(0) = 2. See:
How to Find a Particular Solution for Differential Equations.

Ordinary differential equations have a first derivative as the highest derivative in their solutions; they may be with or without an initial condition. See: How to Solve an Ordinary Differential Equation

Second order differential equations have a second derivative as the highest derivative in their solution. How to Find a Solution to a Second Order Differential Equation.

Definitions

Bessel Functions
What is the Ornstein-Uhlenbeck Process?

Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!