What is the method of undetermined coefficients?

A method for finding a particular solution by choosing a trial form with unknown coefficients, substituting it into the differential equation, and solving for those coefficients.

What are the two main suitability conditions?

The equation has constant coefficients, and its forcing term is built from constants, polynomials, exponentials, sines, cosines, or finite sums and products of these.

What is the complementary solution $y_c$?

The general solution of the associated homogeneous equation, found by setting the forcing term to zero.

How is the general solution of a nonhomogeneous equation formed?

Add a particular solution to the complementary solution: $y=y_c+y_p$.

How does superposition handle a sum of forcing terms?

Find a particular solution for each forcing term separately and add them. If $g(x)=h(x)+w(x)$, then $y_{p_1}+y_{p_2}$ is a particular solution.

Differential equationsLesson 1 of 1 in the course Higher-order linear equations

Undetermined Coefficients: Trial Forms and Resonance

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

The method of undetermined coefficients is a practical way to find a particular solution of certain nonhomogeneous linear ordinary differential equations. It works by choosing a suitable trial form, substituting it into the equation, and solving for unknown coefficients.

A linear nonhomogeneous equation can be written in the form

$y^{(n+1)}+\sum_{i=0}^{n}c_i(x)y^{(i)}=g(x).$

All flashcards

What is the method of undetermined coefficients?: A method for finding a particular solution by choosing a trial form with unknown coefficients, substituting it into the differential equation, and solving for those coefficients.
What are the two main suitability conditions?: The equation has constant coefficients, and its forcing term is built from constants, polynomials, exponentials, sines, cosines, or finite sums and products of these.
What is the complementary solution $y_c$ ?: The general solution of the associated homogeneous equation, found by setting the forcing term to zero.
How is the general solution of a nonhomogeneous equation formed?: Add a particular solution to the complementary solution: $y=y_c+y_p$ .
How does superposition handle a sum of forcing terms?: Find a particular solution for each forcing term separately and add them. If $g(x)=h(x)+w(x)$ , then $y_{p_1}+y_{p_2}$ is a particular solution.

Quiz questions

For forcing term $\sin(\beta x)$ , which trial form includes the terms needed under differentiation?Answer: $A\cos(\beta x)+B\sin(\beta x)$
After substituting a trial function into a differential equation, how are its unknown coefficients typically determined?Answer: Collect like terms and equate corresponding coefficients.
If $h(x)$ and $w(x)$ have particular solutions $y_{p_1}$ and $y_{p_2}$ , what is a particular solution for forcing $h(x)+w(x)$ ?Answer: $y_{p_1}+y_{p_2}$
If a trial form overlaps with a homogeneous solution through a simple characteristic root, what adjustment removes the overlap?Answer: Multiply the entire trial form by $x$ .
If overlap is caused by a characteristic root of multiplicity $m$ , what factor is used to adjust the trial form?Answer: Multiply the trial form by $x^m$ .

Based on the Wikipedia article Method of undetermined coefficients, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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