MATH1012 - Lecture 29

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MATH1012 - Lecture 28
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Nonhomogeneous, Constant Coefficient

• To solve the 2nd order linear DE with constant coefficients: ($g(x)\ne 0)$ $\frac{d^{2}y}{d{x}^2}+p\frac{dy}{dx}+qy=g(x)$
  1. Find solution $y_c(x)$ of associated homogeneous DE
     • $\frac{d^{2}y}{d{x}^2}+p\frac{dy}{dx}+qy=0$
     • Called the complementary function
  2. Then find any particular solution $y_p(x)$ of the original nonhomogeneous differential equation
     • See below
  3. The general solution of the nonhomogeneous differential equation is $y(x)=y_c(x)+y_p(x)$

Method of Undetermined Coefficients

There is an easy way to find $y_p(x)$ if $g(x)$ is in a certain form:

• If $g(x)$ is the sum of different expressions in the left column of the table then $y_p(x)$ can be taken as the sum of the corresponding solutions
• Solve $y_p(x)$ by substituting it into the original DE

One Thing to Watch Out For

If $g(x)$ has a term which appears in the complementary function ($y_c(x)$) then the suggested guess won’t work. We should multiply our original guess by $x$ and try again.

What About Other Right-Hand Sides?

• If $g(x)$ isn’t in our table, we can use variation of parameters
  • e.g. $g(x)=\tan x$, $g(x)=\ln (x)$
• This method is in the Unit Reader but is not examinable

Affine Spaces

• An affine space is a translated vector space
• The $n^{th}$ order inhomogeneous differential equation will have a general solution that is an affine function space
  • This content is not examinible

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Laplace Transforms

• Let $f(t)$ be defined for all $t\ge 0$, the Laplace transform ($LT$) of $f(t)$ is: $\mathcal{L}(f)(s)={\int }_0^{\infty }{e}^{-st}f(t)dt$
  • i.e. $f(t)\to [LT]\to s(t)$
  • It is an improper integral that must be convergent for a solution
• We use the notation:
  • $F(s)$ or just $\mathcal{L}(f)$ for $\mathcal{L}(f)(s)$
  • $t$ is the variable for $f(t)$ and $s$ is the variable for $F(s)$
• If $F(s)$ is the Laplace transform of $f(t)$, then $f(t)$ is the inverse Laplace transform ($ILT$) of $F(s)$
  • There is no formula to compute the inverse Laplace transform
  • Instead we refer to a table of Laplace transforms

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MATH1012 - Lecture 30
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