Table of Contents
What is Variation in Maths?
Variation is a measure of how much one quantity changes in relation to another quantity. It is used to find out how much two quantities vary from each other.
Variation of Parameters
The variation of parameters method is a numerical technique used to solve differential equations. It is a variation of the Runge-Kutta method.
The technique works by varying the parameters of the differential equation in order to find a solution that best matches the given data.
Method of Variation of Parameters
The method of variation of parameters is a technique used to find the solution to a differential equation. The method uses the initial conditions of the equation to find a series of solutions. Each solution is a function of the parameter being varied.
Two Methods in Variation of Parameters
The first method is the direct substitution method. In this method, the parameter is varied and the resulting equation is solved.
The second method is the variation of parameters method. In this method, the parameter is varied and the resulting equations are solved simultaneously.
Solutions to Variation of Parameters
The solutions to variation of parameters are found by taking the derivative of the original equation with respect to each parameter and setting it equal to zero. This will result in a set of equations that can be solved for the unknowns in the equation.
1. First, take the derivative of the original equation with respect to time.
d[itex]\left(\frac{dx}{dt}\right)[/itex]
= [itex]\frac{d}{dt}x[/itex]
= [itex]\frac{d}{dt}(x^{2}+y^{2})[/itex]
Next, set this derivative equal to zero and solve for time.
[itex]\frac{d}{dt}x^{2}+y^{2}[/itex]= 0
= [itex]\frac{-2x}{y^{2}}[/itex]
2. First, take the derivative of the original equation with respect to x.
d[itex]\left(\frac{dx}{dt}\right)[/itex]
= [itex]\frac{d}{dt}x[/itex]
= [itex]\frac{d}{dt}(x^{2}+y^{2})[/ite
Solved Example using Variation of Parameter Formula
The displacement of a harmonic oscillator is given by
x(t) = A sin(ωt + φ)
where A is the amplitude, ω is the angular frequency, and φ is the phase.
What is the displacement at time t = 0?
x(0) = A sin(ω0t + φ)
x(0) = A sin(ω0t + φ)
