MAT 274 Test 1 review

The test covers sections 1.1-1.3, 2.1-2.5, 3.1-3.5. NO CALCULATORS are allowed on the test. You need to provide only rough sketches of your function. Just be careful that the initial value and the basic behavior of the function is well shown.

1.1 Understand what differential equations are, what their solution is. Be able to sketch a direction field and know how to find an equilibrium solution on it. Be able to write a differential equation to the simple examples from this section.

1.2 Understand what an initial value problem is, what is the difference between general solution and particular solution. Know what integral curves are. Be able to solve the type of differential equations that are in this section.

1.3 Be able to categorize differential equations whether they are linear or non-linear, ordinary or partial differential equations and be able to determine their order. Know how to verify if a given function is a solution to a given differential equation.

2.1 Find general and specific (implicit/explicit) solutions for linear, first order differential equations using the method of integrating factor. Remember to determine for which interval the solution you came up with is valid.

2.2 Find general and specific (implicit/explicit) solutions for linear, first order differential equations if the variables are separable. Remember to determine for which interval the solution you came up with is valid.

2.3 Be able to set up differential equations that are similar to example 1 and example 3.

2.4 Understand the fundamental theorem about the existence and uniqueness of the solution to linear first order differential equations. Know the difference between linear and non-linear differential equations and their solutions. Be able to apply Bernoulli's equation to reduce a non-linear equation into a linear equation for some special cases.

2.5 Autonomous equations and population dynamics. Understand how to find equilibriums and how to classify them according to their stability. Be able to predict qualitative properties of solutions such as their asymptotic behavior and critical values that separate solutions without solving the equations. Also, be able to solve these types of differential equations using separation of variables and partial fractions integrating techniques.

3.1 Be able to recognize linear, homogeneous differential equations with constant coefficients. Know how to find the characteristic equation, how to find the general solution and how to use the initial values to find the particular solution. Be able to roughly SKETCH the graph of the solution and describe its behavior for increasing t.

3.2 Understand the fundamental theorem about the existence and uniqueness of the solution to linear second order differential equations and the principle of superposition. Be able to find the Wronskian of two functions. Know what is a fundamental set of solutions is to a given differential equation.

3.3 Be able to decide if two given functions are linearly independent. Understand the statement of Abel's theorem and use it to find the Wronskian of two functions.

3.4 Review complex numbers and Euler's formula. After solving the characteristic equation, know how to find the general solution and how to use the initial values to find the particular solution. Be able to roughly SKETCH the graph of the solution and describe its behavior for increasing t.

3.5 After solving the characteristic equation, know how to find the general solution and how to use the initial values to find the particular solution. Be able to roughly SKETCH the graph of the solution and describe its behavior for increasing t. Understand and be able to apply the method of reduction of order to find a second solution if one is given to a second order linear differential equation.