# Math 256 Study Guide

**Chapter 1: Introduction
**

• Draw direction fields for differential equations of the form y' = f(t, y) (1.1).

• Understand the definition of general solution and particular solution (1.2, 1.3).

• Be able to show given functions are solutions to differential equations (1.3).

• Classify differential equations by order and linearity (1.3).

**Chapter 2: First order differential equations**

• Solve first order linear differential (y' + p(t)y = g(t)) using integrating factors

(2.1).

• Write solutions in terms of definite integrals if needed (2.1).

• Solve first order nonlinear equations by separation. Separable differential equations are of the form

• Use differential equations to answer modeling questions (Tank problems and interest problems in 2.3).

• Understand when solutions are guaranteed to exist and be unique for linear and nonlinear equations

(2.4).

• Understand what types of behavior solutions may exhibit if solutions are not unique (2.4).

• Graph solutions to autonomous equations (y' = f(y)) using the graph of "y versus f(y)" and without

solving for explicit solutions (2.5).

• Understand equilibrium solutions and determine if they are stable, unstable or semi-stable (2.5).

• Determine if a differential equation is exact and find solutions to exact equations. Exact equations are

of the form M(x, y) + N(x, y)y' = 0 where (2.6).

• Approximate solutions to differential equations of the form y' = f(t, y) using Euler's method. Euler's

method uses the equation of a tangent line and the process given by the recursive formula

where h is a fixed differential (2.7).

Chapter 3: Second order differential equations

Chapter 3: Second order differential equations

• Find general and particular solutions to second order, homogeneous, linear equations with constant

coefficients (ay'' + by' + cy = 0) (3.1, 3.3, 3.4). In order to find solutions we look at the characteristic

equation ar

^{2}+br +c = 0. The roots of the polynomial determine the form of the solutions. The three

cases are:

1. , both real .

2. are complex .

3. , both real.

• Compute the Wronskian of two functions and understand the connection to fundamental sets of solu-

tions to second order linear homogenous equations (3.2).

• Understand when solutions are guaranteed to exist and be unique for linear second order differential

equations (3.2).

• Find particular solutions to nonhomogeneous equations using the method of undetermined coefficients

(3.5). The types of functions covered are:

1. Exponentials : .

2. Sine and cosine: .

3. Polynomials: .

and all combinations of products and sums of these functions. The variable s = 0, 1, or 2 depending

on the roots of the characteristic equation.

• Find general solutions to nonhomogeneous equations using solutions to homogeneous equations (y =

(3.5).

**Chapter 4: Higher order differential equations**

• Understand how the techniques used to study second order linear differential equations extend to higher

order differential equations (4.1).

• Compute the Wronskian of three or four functions (4.1).

• Find general and particular solutions to higher order, homogeneous, linear equations with constant

coefficients. There equations are of the form (4.2).

• Find roots to characteristic polynomials using long division .

• Find roots to characteristic polynomials using Euler's equation (4.2).

• Find particular solutions to nonhomogeneous equations using the method of undetermined coefficients

for higher order differential equations (4.3).

**Chapter 7: Systems of differential equations**

• Solve systems of two first order differential equations by rewriting the system as single second order

equation (7.1).

• Write a system of differential equations as a single higher order linear differential equation (7.1).

• Write a linear differential equation (of order ≥ 2) as a system of first order differential equations (7.1).

• Add , multiply and scale matrices (7.2).

• Write systems of differential equations in matrix form (7.3).

• Solve a given system of linear equations using row reduction of matrices (7.3).

• Find eigenvalues and eigenvectors of square matrices (7.3).

• Write solutions to differential equations in vector form (7.2, 7.4).

• Compute the Wronskian of a system of solutions (7.4).

• Find solutions to systems of differential equation using eigenvectors and eigenvalues (7.5).

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