Find transient terms in this general solution to a differential equation, if there are any y=(x+C)(frac{x+2}{x-2})

Rui Baldwin 2021-09-15 Answered
Find transient terms in this general solution to a differential equation, if there are any
\(\displaystyle{y}={\left({x}+{C}\right)}{\left({\frac{{{x}+{2}}}{{{x}-{2}}}}\right)}\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

unett
Answered 2021-09-16 Author has 10902 answers

\(\displaystyle{\left({x}+{C}\right)}{\left({\frac{{{x}+{2}}}{{{x}-{2}}}}\right)}={\frac{{{x}^{{2}}}}{{{x}-{2}}}}+{\frac{{{\left({2}+{C}\right)}{x}}}{{{x}-{2}}}}+{\frac{{{2}{C}}}{{{x}-{2}}}}\)
Then take the limits of each of these three terms as \(x\rightarrow\infty\) and see which(if any) approach 0. All three are rational expressions, so it is clear that only \(\displaystyle{\frac{{{2}{C}}}{{{x}-{2}}}}\) is transient term.

Have a similar question?
Ask An Expert
32
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-11-05
Find the general solution for the following differential equation.
\(\displaystyle{\frac{{{d}^{{{3}}}{y}}}{{{\left.{d}{x}\right.}^{{{3}}}}}}-{\frac{{{d}^{{{2}}}{y}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}-{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{y}={x}+{e}^{{-{x}}}\).
asked 2021-10-23
Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution.
\(\displaystyle{\cos{{x}}}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}+{\left({\sin{{x}}}\right)}{y}={1}\)
asked 2021-11-12
Find the general solution of the given differential equation.
\(\displaystyle{\frac{{{d}{r}}}{{{d}\theta}}}+{r}{\sec{{\left(\theta\right)}}}={\cos{{\left(\theta\right)}}}\)
Give the largest over which the general solution is defined.
asked 2021-11-11
Find the general solution of the given differential equation.
\(\displaystyle{y}^{{{\left({6}\right)}}}-{y}{''}={0}\)
asked 2021-11-14
Find the general solution of the given differential equation.
\(\displaystyle{y}^{{{\left({6}\right)}}}+{y}={0}\)
asked 2021-10-04
Find the general solution of the given differential equation. \(\displaystyle{y}”+{y}'−{6}{y}={12}{e}^{{{3}{t}}}+{12}{e}^{{−{2}{t}}}\)
asked 2021-10-18
Find the general solution of the given differential equation.y” + 2y' + y = 2e−t

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...