A general method to find particular solutions for any continuous forcing function.
Variation of parameters finds y_p=-y₁∫(y₂f/W)dx+y₂∫(y₁f/W)dx with W the Wronskian, valid for any continuous forcing. It is more general but more laborious than undetermined coefficients.
Use it for y''+y=tan x where guessing fails.
Variation of parameters works for any continuous forcing, so it is the general tool when simple guesses fail.
A general method to find particular solutions for any continuous forcing function.
Variation of parameters works for any continuous forcing, so it is the general tool when simple guesses fail.
The constants in the complementary solution are allowed to vary with time, and that variation is solved for using the Wronskian.
Usually yes, because it needs integrals. But it works in every case where the forcing is continuous.
Typical applications include finding the response of a beam to an arbitrary load history, solving a circuit driven by a non-standard waveform and deriving the response of a vibrating system to an impulse.
Study "Undetermined coefficients" first (topic 64 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
Needs two homogeneous solutions. Wronskian must be non-zero. Works with variable coefficients.
Use it for y''+y=tan x where guessing fails.
The next topic is "Damping and resonance", which builds directly on this one.
It is classed as Intermediate in Engineering Mathematics (topic 65 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.
Want to practise? Open the interactive course for quick checks, quizzes, simulations and progress tracking.