12.4: Summary
- Page ID
- 121149
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Given a function, we can check whether it is a solution to a differential equation by performing the appropriate differentiation and algebraic simplification.
- Solutions to differential equations in which there is no change at all ("constant solutions") are referred to as steady states.
- The differential equations
\[\frac{d y}{d t}=a-b y, \quad y(0)=y_{0} \nonumber \]
has a steady state solution \(y=a / b\).
- If we define the deviation from steady state, \(z(t)=y(t)-\frac{a}{b}\), we get a decay equation for \(z(t)\) that has exponentially decreasing solutions provided \(b>0\). This says that the eviation from steady state always decrease over time.
- The resulting solution for \(y(t)\) is
\[y(t)=\frac{a}{b}-\left(\frac{a}{b}-y_{0}\right) e^{-b t} . \nonumber \]
- For some differential equations, it is not always possible to determine an analytic solution (explicit formula). Numerical solutions can be found using Euler’s method, and serve as an approximate solution.
- Euler’s method takes a known initial value \(y_{0}\) and uses the iteration scheme:
\[y_{k+1}=y_{k}+f\left(y_{k}\right) \Delta t . \nonumber \]
to generate successive values of \(y_{k}\) that approximate the solution at time points \(t_{k}=k \Delta t\)
- Applications considered in this chapter included:
- height of water draining out of a cylindrical container (verifying a solution to a differential equation);
- Newton’s law of cooling (described by a linear differential equation);
- growth of the radius of a cell;
- the accumulation of greenhouse gasses in the atmosphere;
- friction and terminal velocity; and
- chemical production and decay.
- Explain why an object at room temperature is at a steady state for Newton’s law of cooling.
- The following graph depicts solution curves to a particular differential equation of the form \(d y / d t=a-b y\).
- Estimate the value that these solution curves are approaching.
- Which solutions are approaching from above or below?
- Consider the following initial value problem:
\[\frac{d y}{d t}=2-4 y, \quad y(0)=4, \nonumber \]
- What value does its solution curve approach?
- Does its solution approach from above or below?
- Why is a large value of \(\Delta t\) not a good idea when using Euler’s method?