Exam 2 Study Guide

• Exam 2 Topics
• Exam 2 Summary
• Review / Study Tips

Exam 2 Topics

• This exam covers weeks 7 through 11 of class. See the Daily Lessons for a complete list of topics.

• Each daily lesson contains a collection of resources you can use to review the lessons. In particular pay attention to:

  • Learning Goals: Each lesson has a list of learning goals. Make sure you know/are able to do everything listed in those goals.

  • Worksheet Solutions: The worksheets have both written and video solutions. These can be useful when going over the worksheets.

  • Notes: The course notes contain the main information needed for each lesson. Most of the notes contain colored boxes which contain key information for that lesson.

Exam 2 Summary

Here is a summary (partial list) of things you should know and be able to do. Use the above links for the full list.

• Know what a system of first order differential equations is.

• Know how to identify linear systems of differential equations and be able to write linear systems in matrix form. Know that if a system is not linear, it is called nonlinear.

• Be able to identify if a linear system is homogeneous or nonhomogeneous.

• Know that solutions to linear homogeneous systems form a vector space. This means solutions are found as follows:

  • Find a set of fundamental solutions.

    $${\overrightarrow{\mathbf{s}}}_1(t),{\overrightarrow{\mathbf{s}}}_2(t),\dots ,{\overrightarrow{\mathbf{s}}}_n(t)$$

  • Fundamental solutions must be linearly independent. The Wronskian can be used to test for linear independence.

  • The general solution is the linear combination of the fundamental solutions.

    $$\overrightarrow{\mathbf{s}}(t)=c_1{\overrightarrow{\mathbf{s}}}_1(t)+c_2{\overrightarrow{\mathbf{s}}}_2(t)+\cdots +c_n{\overrightarrow{\mathbf{s}}}_n(t)$$

• Know that fundamental solutions to linear homogeneous systems with constant coefficients are found using the eigenvalues and eigenvectors of the matrix $A$. Know the three cases for the eigenvalues and eigenvectors.

  • Two Distinct Real Eigenvalues ${\lambda }_1\ne {\lambda }_2$ with eigenvectors ${\overrightarrow{\mathbf{v}}}_1$ and ${\overrightarrow{\mathbf{v}}}_2$:

    $$\begin{array}{rl}{\overrightarrow{\mathbf{s}}}_1(t)& =e^{{\lambda }_{1}t}{\overrightarrow{\mathbf{v}}}_1\\ {\overrightarrow{\mathbf{s}}}_2(t)& =e^{{\lambda }_{2}t}{\overrightarrow{\mathbf{v}}}_2\end{array}$$

  • Two Non-Real Eigenvalues ${\lambda }_{1,2}=\mu \pm i\nu$ with eigenvectors ${\overrightarrow{\mathbf{v}}}_{1,2}=\overrightarrow{\mathbf{u}}\pm i\overrightarrow{\mathbf{w}}$:

    $$\begin{array}{rl}{\overrightarrow{\mathbf{s}}}_1(t)& =e^{\mu t}(\cos (\nu t)\overrightarrow{\mathbf{u}}-\sin (\nu t)\overrightarrow{\mathbf{w}})\\ {\overrightarrow{\mathbf{s}}}_2(t)& =e^{\mu t}(\sin (\nu t)\overrightarrow{\mathbf{u}}+\cos (\nu t)\overrightarrow{\mathbf{w}})\end{array}$$

  • One Repeated Real Eigenvalue ${\lambda }_1={\lambda }_2=\lambda$:

    • If $\mathbf{A}=\lambda \mathbf{I}$ is diagonal then:

      $$\begin{array}{rl}{\overrightarrow{\mathbf{s}}}_1(t)& =e^{\lambda t}\left[\begin{array}{c}1\\ 0\end{array}\right]\\ {\overrightarrow{\mathbf{s}}}_2(t)& =e^{\lambda t}\left[\begin{array}{c}0\\ 1\end{array}\right]\end{array}$$

    • If $\mathbf{A}=\lambda \mathbf{I}$ is not diagonal then:

      $$\begin{array}{rl}{\overrightarrow{\mathbf{s}}}_1(t)& =e^{\lambda t}\overrightarrow{\mathbf{v}}\\ {\overrightarrow{\mathbf{s}}}_2(t)& =(t\overrightarrow{\mathbf{v}}+\overrightarrow{\mathbf{w}})e^{\lambda t}\end{array}$$

      where $\overrightarrow{\mathbf{v}}$ is an eigenvector and $\overrightarrow{\mathbf{w}}$ is a generalized eigenvector that satisfies the matrix equation

      $$(\mathbf{A}-\lambda \mathbf{I})\overrightarrow{\mathbf{w}}=\overrightarrow{\mathbf{v}}$$

• Be able to use the eigenvalues and eigenvectors to describe behavior of solutions in the phase plane and determine the stability of the equilibrium solution at the origin.

• Know what a second order differential equation is and be able to identify if a second order differential equation is linear or nonlinear.

• Be able to identify if a linear second order differential equation is homogeneous or nonhomogeneous.

• Be able to write a linear second order differential equation as a system of two first order differential equations by using the substitution $\overrightarrow{\mathbf{s}}=\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]=\left[\begin{array}{c}y\\ y^{\prime }\end{array}\right]$

• Know that solutions to linear homogeneous second order differential equations form a vector space. This means that solutions are found as follows:

  • Find a set of two fundamental solutions:

    $$y_1(t)\text{ and }{y}_2(t)$$

  • Fundamental solutions are linearly independent, so their Wronskian is non-zero.

    $$W(y_1,y_2)=\left|\begin{array}{cc}{y}_1& y_2\\ y_1^{\prime }& y_2^{\prime }\end{array}\right|$$

  • The general solution is a linear combination of the fundamental solutions:

    $$y(t)=c_1{y}_1(t)+c_2{y}_2(t)$$

• Be able to find fundamental solutions to second order linear homogeneous differential equations with constant coefficients:

  $$a{y}^{″}+b{y}^{\prime }+cy=0$$

  • First solve the characteristic equation:

    $$a{\lambda }^2+b\lambda +c=0$$

  • The fundamental solutions depend on the roots of the characteristic equation:

    • Two real roots ${\lambda }_1\ne {\lambda }_2$:

      $$\begin{array}{rl}{y}_1(t)& =e^{{\lambda }_{1}t}\\ y_2(t)& =e^{{\lambda }_{2}t}\end{array}$$

    • One repeated real root ${\lambda }_1={\lambda }_2=\lambda$:

      $$\begin{array}{rl}{y}_1(t)& =e^{\lambda t}\\ y_2(t)& =t{e}^{\lambda t}\end{array}$$

    • Two non real roots ${\lambda }_{1,2}=\mu \pm i\nu$:

      $$\begin{array}{rl}{y}_1(t)& =e^{\mu t}\cos (\nu t)\\ y_2(t)& =e^{\mu t}\sin (\nu t)\end{array}$$

• Know that solutions to linear nonhomogeneous second order differential equations are of the form

  $$y(t)=y_h(t)+y_p(t)$$

  where $y_h(t)$ is the general solution to the related homogeneous equation and $y_p(t)$ is any particular solution to the nonhomogeneous equation.

• Be able to find particular solutions to linear nonhomogeneous equations using either of the two methods:

  • Undetermined Coefficients which guesses the form of the solution and checks it trying to find values of unknown constants to make it work.

  • Variation of Parameters which finds the particular solution using integration formulas. The particular solution to a DE written in the form

    $$y^{″}+p(t)y^{\prime }+q(t)y=g(t)$$

    is of the form

    $$y_p(t)=u_1(t)y_1(t)+u_2(t)y_2(t)$$

    where $y_1(t)$ and $y_2(t)$ are fundamental solutions and $u_1(t)$ and $u_2(t)$ are found using the following formulas:

    $$\begin{array}{rl}{u}_1(t)& =-\int \frac{y_2(t)g(t)}{W(y_1,y_2)}dt\\ u_2(t)& =\int \frac{y_1(t)g(t)}{W(y_1,y_2)}dt\end{array}$$

• Be able to use systems of differential equations to work with and answer questions about tank mixing problems with a system of two interconnected tanks.

• Be able to use second order differential equations to answer questions about both unforced spring mass systems and forced spring mass systems.

Review / Study Tips

Here are some tips to help you study for the exam in order of importance:

• Review the old quizzes. Rework any problem you had trouble with.

• Read over all the learning goals on the Daily Lessons page.

• Complete the three exam review worksheets (these worksheets contain old exam questions).

• Review the worksheets and their solutions.

• Review the WeBWorK assignments. Rework the problems you had trouble with.