English | Español

Try our Free Online Math Solver!

Online Math Solver

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


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


• 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 ar2 +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).

Prev Next