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.

    \[\vec{\mathbf{s}}_1(t), \vec{\mathbf{s}}_2(t), \dotsc, \vec{\mathbf{s}}_n(t)\]

  • 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.

    \[\vec{\mathbf{s}}(t) = c_1 \vec{\mathbf{s}}_1(t) + c_2 \vec{\mathbf{s}}_2(t) + \dotsb + c_n \vec{\mathbf{s}}_n(t)\]

• 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 \( \lambda_1 \neq \lambda_2 \) with eigenvectors \( \vec{\mathbf{v}}_1 \) and \( \vec{\mathbf{v}}_2 \):

    \[\begin{align*} \vec{\mathbf{s}}_1(t) &= e^{\lambda_1 t} \vec{\mathbf{v}}_1 \\ \vec{\mathbf{s}}_2(t) &= e^{\lambda_2 t} \vec{\mathbf{v}}_2 \end{align*}\]

  • Two Non-Real Eigenvalues \( \lambda_{1,2} = \mu \pm i \nu \) with eigenvectors \( \vec{\mathbf{v}}_{1,2}=\vec{\mathbf{u}} \pm i \vec{\mathbf{w}} \):

    \[\begin{align*} \vec{\mathbf{s}}_1(t) &= e^{\mu t} \left( \cos(\nu t)\vec{\mathbf{u}} - \sin(\nu t)\vec{\mathbf{w}} \right) \\ \vec{\mathbf{s}}_2(t) &= e^{\mu t} \left( \sin(\nu t)\vec{\mathbf{u}} + \cos(\nu t)\vec{\mathbf{w}} \right) \end{align*}\]

  • One Repeated Real Eigenvalue \(\lambda_1 = \lambda_2 = \lambda\):

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

      \[\begin{align*} \vec{\mathbf{s}}_1(t) &= e^{\lambda t} \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\ \vec{\mathbf{s}}_2(t) &= e^{\lambda t} \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{align*}\]

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

      \[\begin{align*} \vec{\mathbf{s}}_1(t) &= e^{\lambda t} \vec{\mathbf{v}} \\ \vec{\mathbf{s}}_2(t) &= (t \vec{\mathbf{v}} + \vec{\mathbf{w}}) e^{\lambda t} \end{align*}\]

      where \( \vec{\mathbf{v}} \) is an eigenvector and \( \vec{\mathbf{w}} \) is a generalized eigenvector that satisfies the matrix equation

      \[(\mathbf{A} - \lambda \mathbf{I}) \vec{\mathbf{w}} = \vec{\mathbf{v}}\]

• 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 \(\vec{\mathbf{s}} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} y \\ y' \end{bmatrix}\)

• 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.

    \[W(y_1,y_2) = \left| \begin{matrix} y_1 & y_2 \\ y'_1 & y'_2 \end{matrix} \right|\]

  • 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:

  \[a y'' + b y' + c y = 0\]

  • 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 \(\lambda_1 \neq \lambda_2\):

      \[\begin{align*} y_1(t) &= e^{\lambda_1 t} \\ y_2(t) &= e^{\lambda_2 t} \end{align*}\]

    • One repeated real root \(\lambda_1 = \lambda_2 = \lambda\):

      \[\begin{align*} y_1(t) &= e^{\lambda t} \\ y_2(t) &= t e^{\lambda t} \end{align*}\]

    • Two non real roots \(\lambda_{1,2} = \mu \pm i \nu\):

      \[\begin{align*} y_1(t) &= e^{\mu t} \cos(\nu t) \\ y_2(t) &= e^{\mu t} \sin(\nu t) \end{align*}\]

• 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

    \[y'' + p(t) y' + q(t) y = g(t)\]

    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:

    \[\begin{align*} u_1(t) &= - \int \frac{y_2(t) g(t)}{W(y_1,y_2)} dt \\ u_2(t) &= \int \frac{y_1(t) g(t)}{W(y_1,y_2)} dt \end{align*}\]

• 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.