Study Guide
Differential equations: Chapter 3
How to use this study guide:
You should understand each ♠ point, by using the suggested resources
or any others. Corresponding exercises are recommended, but not required. Answers for exercises not in the book are given at the end of this
guide.
Section 3.1
♠ Know the form of a general linear DE of 2nd order. Also know what
it means for the equation to be homogeneous, and what it means to be
a constant coefficient equation.
Reference: the definitions are on p.137-139
♠ Know how to derive the characteristic equation for 2nd order linear
DE with constant coefficients. Also know how to find the general solution
of a 2nd order linear DE with constant coefficients when the characteristic
equation has real roots.
Reference: p.140 and example 2-4 on p.141-143
Exercise: 1-16
Section 3.2
♠ Know three methods for solving a 2 × 2 system of linear equations:
• Adding or subtracting equations
• Substitution
• Cramer’s rule
Reference: For Cramer’s rule you can read
Wikipedia: http://en.wikipedia.org/wiki/Cramer’s_rule#Explicit_
formulas_for_small_systems
Coolmath (understandable by middle schoolers): http://www.coolmath.
com/algebra/14-determinants-cramers-rule/01-determinants-cramers-rule-2x2-01.
htm
♠ Understand the basics of complex numbers:
• graphing a complex number on the complex plane
• real and imaginary parts of a complex number
• addition, subtraction
• multiplication (don’t need division)
• absolute value of a complex number (also called modulus)
• conjugate
• rectangular and polar form of a complex number
• exponential form of a complex number
Reference: There are many out there–one with videos is Khan Academy:
1
2
https://www.khanacademy.org/math/precalculus/imaginary_complex_
precalc
Exercise: there are interactive exercises on the Khan Academy page
♠ Know the existence and uniqueness theorem for 2nd order linear
equations.
Reference: Theorem 3.2.1 on p.146
Exercise: 7-12
♠ What is the principle of superposition?
Reference: Theorem 3.2.2 on p.147
Exercise: 13
♠ What is the Wronskian determinant (also called the Wronskian) of
two functions? What is the significance of the Wronskian of two solutions
of a 2nd order linear DE?
Reference: Equation (9) on p.148 tells you how to compute the Wronskian, and Theorems 3.2.3 and 3.2.4 explain what it tells you.
♠ Know what a fundamental set of solutions to a 2nd order linear DE
is.
Reference: bottom of p.150. The significance of all this is that if the
Wronskian of two solutions y1 and y2 is not equal to zero at any point,
then it is never zero for any t (as long as the equation has continuous
coefficients). Therefore, y1 and y2 form a fundamental set of solutions.
Exercise: 24-27
♠ Know what Abel’s theorem says and why it is significant.
Reference: Theorem 3.2.6 on p.153. The significance is that it lets you
compute the Wronskian of the solutions y1 and y2 of a 2nd order linear
DE up to a multiplicative constant without knowing y1 and y2 themselves.
Section 3.3
♠ Know how to find the general solution of 2nd order linear DE when
the roots of the characteristic equation are complex. Also know how to
solve IVP involving such equations, and what typical graphs of solutions
look like.
Reference: Equation (24) on p.161, and examples 2 and 3.
Exercise: 7-22
Section 3.4
♠ Know how to find the general solution of 2nd order linear DE when
the characteristic equation has a repeated root. Also know how to solve
3
IVP involving such equations, and what typical graphs of solutions look
like.
Reference: Equation (18) on p.168, and example 2 on p.169.
Example: 1-14
♠ Know how to use the reduction of order method to find a fundamental set of solutions to a 2nd order linear DE (with possibly variable
coefficients) when one solution is known.
Reference: bottom of p.170 and example 3.
Exercise: 23-30
4
Answers to selected exercises