Glossary
Constant of Integration (+C)
An arbitrary constant introduced when performing indefinite integration, representing the family of antiderivatives for a given function.
Example:
When integrating ∫2x dx, we get x² + C, where C accounts for any vertical shift of the antiderivative.
Domain Restrictions
Limitations on the valid input values for which a solution to a differential equation is defined, often arising from mathematical constraints like division by zero or physical limitations.
Example:
For the solution y = ln|x-2|, there's a domain restriction that x cannot equal 2 because the natural logarithm is undefined there.
General Solution
A general solution to a differential equation is a family of functions that satisfies the equation, containing an arbitrary constant (C). It represents infinitely many possible solutions.
Example:
The general solution to dy/dx = 2x is y = x² + C, representing a family of parabolas shifted vertically.
Initial Conditions
A specific point (x₀, y₀) provided for a differential equation, used to determine the unique value of the constant of integration and thus find a particular solution.
Example:
For a population growth model dP/dt = kP, the initial condition P(0) = 100 tells us the starting population is 100 individuals at time t=0.
Particular Solution
A particular solution is a unique function that satisfies a differential equation and passes through a specific point, determined by using an initial condition to solve for the constant of integration.
Example:
Given dy/dx = 2x and the initial condition y(1)=3, the particular solution is y = x² + 2, found by solving for C.
Separation of Variables
A technique used to solve certain differential equations by algebraically rearranging the equation so that all terms involving one variable are on one side and all terms involving the other variable are on the opposite side.
Example:
To solve dy/dx = x/y, you would use separation of variables to get y dy = x dx before integrating.
Singularities
Points or values where a mathematical expression becomes undefined, such as division by zero, leading to an undefined solution for a differential equation at those points.
Example:
In the differential equation dy/dx = 1/(x-3), x=3 is a singularity because the derivative is undefined there.