ezLecture: An Introduction to Differential Equations Using Classical Mechanics
NOTES
Learning Goals
1. To identify different types of differential equations.
2. To develop techniques to solve differential equations (separate and integrate/integration
factors/guess and check).
Our text of reference will be Boaz’s third edition of Mathematical Methods in Physical Sciences, Chapter
8, Sections 1-3.
1. Identify the types of differential equations
A) The order of a differential equation is the order of the highest derivative in the equation. The
following differential equations are both second order.
2
𝑑 𝑟⃗
𝐹⃗ = 𝑚 𝑑𝑡 2
𝐿
𝑑2 𝐼
𝑑𝑡 2
+𝑅
𝑑𝐼
𝑑𝑡
𝐼
𝐶
+ =0
B) These two equations are ordinary, as they don’t contain any partial derivatives.
C) Another important distinction is that these differential equations are linear, which means that they
are of the form:
𝑑𝑦
𝑑2 𝑦
𝑎0 𝑦 + 𝑎1 𝑑𝑡 + 𝑎2 𝑑𝑡 2 + ⋯ = 𝑏,
where 𝑦 is the dependent variable, 𝑡 is the independent variable, and the a’s and b’s are constants or
functions of 𝑡.
D) If 𝑏 = 0, then the differential equation is said to be homogeneous, and if 𝑏 is a function of 𝑡 or a
constant, then the equation is said to be inhomogeneous.
2. Develop techniques to solve differential equations.
A) We want to develop techniques to find solutions to differential equations. A general solution to the
differential equation will represent a family of curves that are all solutions to the differential equation.
For linear differential equations of order n, the general solution will contain n independent arbitrary
constants. For example, a second order linear differential equation should contain 2 independent
arbitrary constants. A particular solution, on the other hand, is the one curve in the family of curves that
satisfies the differential equation and some other requirements, such as the boundary conditions.
B) The general solution to an inhomogeneous differential equation will contain a particular solution that
solves the differential equation and the complementary solution, which is the general solution to the
homogeneous version of the differential equation.
𝑦 = 𝑦𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 + 𝑦𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
C) Separate and Integrate – If the differential equation is separable, you can separate the independent
and dependent variables on to opposite sides of the equation, then integrate both sides. Use the
boundary conditions to solve for the constants and find the particular solution.
𝑑𝑦
𝑑𝑥
= 𝑓(𝑥).
𝑑𝑦 = 𝑓(𝑥)𝑑𝑥
∫ 𝑑𝑦 = ∫ 𝑓(𝑥)𝑑𝑥
𝑦 = ∫ 𝑓(𝑥)𝑑𝑥
D) Use an Integration Factor – If the equation is a first order, linear equation you can find the following
integration factor and use it to get the solution.
𝑑𝑦
+ 𝑎𝑦
𝑑𝑡
𝑒 ∫ 𝑎𝑑𝑡 =
= 𝑏, where 𝑎, 𝑏 are constants or functions of t.
𝑒𝐼
𝑦 = 𝑒 −𝐼 ∫ 𝑏𝑒 𝐼 𝑑𝑡 + 𝐶[𝑒 −𝐼 ], where 𝐶 is a constant.
E) Guess and Check – First guess the form the solution should take, and then check to see if the solution
is correct by plugging in the solution and its derivatives into the differential equation.