Method of Undetermined Coefficients

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Table of Contents

• Method of Undetermined Coefficients
  • Key Definitions
  • Method of Undetermined Coefficients
    • Selecting a Trial Function
      • $g(t)=e^{rt}$
        • $y_p(t)=a{e}^{rt}$
      • $g(t)=A\cos \omega t+B\sin \omega t$
        • $y_p(t)=a\cos \omega t+b\sin \omega t$
      • $g(t)=P(t)$
        • $y_p(t)=p_0(t)$
      • $g(t)=P(t)\cos \omega t$ or $g(t)=P(t)\sin \omega t$
        • $y_p(t)=p_0(t)\cos \omega t+p_1(t)\sin \omega t$
      • $g(t)=e^{rt}\cos \omega t$ or $g(t)=e^{rt}\sin \omega t$
        • $y_p(t)=e^{rt}(a\cos \omega t+b\sin \omega t)$
      • $g(t)=e^{rt}P(t)\cos \omega t$ or $g(t)=e^{rt}P(t)\sin \omega t$
        • $y_p(t)=e^{rt}(p_0(t)\cos \omega t+p_1(t)\sin \omega t)$
        • $t{y}_p(t)\text{or}{t}^2{y}_p(t)$
    • Attempting a Solution
      • $y_P(t)=g(t)$
  • Superposition Principle
    • $y^{″}+p{y}^{\prime }+qy=\alpha f(t)+\beta g(t)$
    • $y(t)=\alpha y_f(t)+\beta y_g(t)$

#UCLA #Y1Q3 #Math33B

Method of Undetermined Coefficients

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Key Definitions

Method of Undetermined Coefficients - used to find particular sol. to if:

• $p,q$ are constant functions

Trial Solution - arbitrary possible solution given by restraints:

• must include forcing term $g(t)$
• must be “closed” (similar) under derivation (e.g. trig funcs.)

Superposition Principle - used to deal with lin. combs. of forcing terms

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Method of Undetermined Coefficients

Given where $p,q$ are constant and forcing term $g(t)$ is “closed” under derivation, we use a trial solution containing an undetermined coeff.

Selecting a Trial Function

The trial solution depends on the forcing term, if $g(t)$ is not a sol.:

1. $g(t)=e^{rt}$

$y_p(t)=a{e}^{rt}$

1. $g(t)=A\cos \omega t+B\sin \omega t$

$y_p(t)=a\cos \omega t+b\sin \omega t$

1. $g(t)=P(t)$

$y_p(t)=p_0(t)$

1. $g(t)=P(t)\cos \omega t$ or $g(t)=P(t)\sin \omega t$

$y_p(t)=p_0(t)\cos \omega t+p_1(t)\sin \omega t$

1. $g(t)=e^{rt}\cos \omega t$ or $g(t)=e^{rt}\sin \omega t$

$y_p(t)=e^{rt}(a\cos \omega t+b\sin \omega t)$

1. $g(t)=e^{rt}P(t)\cos \omega t$ or $g(t)=e^{rt}P(t)\sin \omega t$

$y_p(t)=e^{rt}(p_0(t)\cos \omega t+p_1(t)\sin \omega t)$

s.t. $A,B,a,b,r,\omega \in \mathbb{R}$ and $P(t),p_0(t),p_1(t)$ are polynomials of the same degree

if $g(t)$ is a sol. use

$t{y}_p(t)\text{or}{t}^2{y}_p(t)$

Attempting a Solution

Set the trial equal to the forcing term and solve for the undetermined coefficient to find that the trial function is a particular solution

$y_P(t)=g(t)$

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Superposition Principle

if $y_f(t)$ is a part. sol. to $y^{\prime }‘+p{y}^{\prime }+qy=f(t)$ and $y_g(t)$ is a part. sol. to $y^{\prime }‘+p{y}^{\prime }+qy=g(t)$, and given:

$y^{″}+p{y}^{\prime }+qy=\alpha f(t)+\beta g(t)$

then the general solution is:

$y(t)=\alpha y_f(t)+\beta y_g(t)$