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\title{Undetermined coefficients}
\date{March 3, 2021}


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\vspace{-.2in}

\thm{
  If $y_p$ is any solution to the differential equation $ y'' + p(x) y' + q(x) y = f(x)$ and $\{y_1, y_2\}$ is a fundamental set of solutions to the complementary equation $y'' + p(x) y' + q(x) y = 0$, then a general solution for the original differential equation is \[ \boxed{ y = y_p +c_1 y_1 + c_2 y_2 }\]
}

\thm{[Superposition]
If $y_{p_1}$ is a particular solution of $ y'' + p(x) y' + q(x) y = f_1(x)$ and $y_{p_2}$ is a particular solution of $y'' + p(x) y' + q(x) y = f_2(x)$, then a particular solution of \[ y'' + p(x) y' + q(x) y = f_1(x) + f_2(x)\] is \[ y_p = y_{p_1} + y_{p_2}\]
}

\noindent \hrulefill

\mtd{
  To find a solution $y_p$ to the differential equation $a y'' + b y' + c y = P(x)$ where $P(x)$ is a polynomial of degree $n$ (and $c \neq 0$):
  set $y_p = A_nx^n + A_{n-1} x^{n-1} + \dots + A_0$ and solve for the undetermined coefficients $A_0, A_1, \dots, A_n$.
}

\eg{
Solve the IVP $y'' + 2y' + y = t - 3$, $y(0)=1$, $y'(0) = -1$.
}

\noindent \hrulefill
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\mtd{
  To find a solution $y_p$ to the differential equation $ a y'' + b y' + c y = ke^{\alpha x}$: set $y_p$ equal to the first of the following that is not a solution to the complementary equation and solve for the undetermined coefficients.
  \begin{enumerate}
  \item $y_p = Ae^{\alpha x}$
  \item $y_p = Axe^{\alpha x}$
  \item $y_p = Ax^2e^{\alpha x}$
  \end{enumerate}
}

\prob{[Completion]
Solve the IVP $y'' + 3y' + 2y = e^{-t}$, $y(0)=1$, $y'(0) = -1$.
}

\noindent \hrulefill

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\mtd{
 To find a solution $y_p$ to the differential equation $ a y'' + b y' + c y = p \cos(\omega x) + q \sin(\omega x)$: set $y_p$ equal to the first of the following that is not a solution to the complementary equation and solve for the undetermined coefficients.
 \begin{enumerate}
 \item $y_p = A \cos (\omega x) + B \sin(\omega x)$
 \item $y_p = A x \cos (\omega x) + B x \sin(\omega x)$
 \end{enumerate}
}

\prob{[Graded]
  Solve the IVP $y''+y = \frac{1}{3}\cos t$, $y(0) = 0$, $y'(0) = 0$.
}

\prob{[Graded]
 Find a particular solution to $y''+y = \frac{1}{3}\cos(2 t)$
}
  
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