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
      • $g(t)=e^{rt}$
        • \(y_p(t)=ae^{rt}\)
      • $g(t)=A\cos\omega t + B\sin\omega t$
        • \(y_p(t)=a\cos\omega t + b\sin\omega t\)
      • $g(t)=P(t)$
        • \(y_p(t)=p_0(t)\)
      • $g(t)=P(t)\cos\omega t$ 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 $g(t)=e^{rt}P(t)\sin\omega t$
        • \(y_p(t)=e^{rt}(p_0(t)\cos\omega t + p_1(t)\sin\omega t)\)
        • \(ty_p(t)\quad\text{or}\quad t^2y_p(t)\)
    • Attempting a Solution
      • \(y_P(t)=g(t)\)
  • Superposition Principle
    • \(y''+py'+qy=\alpha f(t)+\beta g(t)\)
    • \(y(t)=\alpha y_f(t) + \beta y_g(t)\)

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

\(y_p(t)=ae^{rt}\)

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

\(ty_p(t)\quad\text{or}\quad t^2y_p(t)\)

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 $y’‘+py’+qy=f(t)$ and $y_g(t)$ is a part. sol. to $y’‘+py’+qy=g(t)$, and given:

\(y''+py'+qy=\alpha f(t)+\beta g(t)\)

then the general solution is:

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