Review for First Order Differential Equations

1 Integration techniques

1.1 Integration by substitution

Example:

Example:

1.2 Integration by parts

Example:

1.3 Integration by partial fractions

2 Existence and uniqueness

2.1 Linear Equations

For initial value problem

if

•the coefficients p(t) and g(t) are both continuous on (a,b), and
•t is in (a, b),

then the initial value problem has a unique solution on the entire (a, b). (Theorem 2.1)

The general procedure to find such an interval (a, b) for the existence of a unique solution
involves the following steps: 1. Put the linear equation in standard form, identify p(t) and
g(t), 2. Determined the domain of p(t) and g(t), in other words, find where p(t) and g(t)
are discontinuous (use the aid of number axis if necessary), 3. In the domain obtained from
step 1 find the interval where t₀ is in.

Example: For initial value problem 1,Find the largest
t-interval on which theorem 2.1 guarantees the existence of a unique solution.

2.2 Nonlinear equations

For initial value problem

if

•f(t, y) and f_y(t, y) are both continuous on the open rectangle R defined by a < t < b
and α < y < β, and

•(t₀, y₀) is in R,

then there is an open t-interval (c, d), contained in (a, b) and containing t₀ (i.e., a ≤ c <
t₀ < d ≤ b), in which there exists a unique solution of the initial value problem. (Theorem
2.2)

Note that: 1. Theorem 2.2 doesn’t give the exact numbers for (c, d), 2. Theorem 2.2
includes theorem 2.1 when applied to linear equations.

3 First order linear differential equations

3.1 Homogeneous equations

General solution is

where c is a constant and is the antiderivative of p(t).

3.2 Nonhomogeneous equations

General solution is

where the first term is a particular solution to the nonhomogeneous equation, and the
section term is the general solution for the homogenous equation.

You should know how to use integrating factor μ(t) = e^P(t) to turn the left hand side
of the differential equation into a single derivative. But in real applications, probably it’s
faster to directly use the above formula to solve the linear nonhomogeneous differential
equations.

Example: Solve
Example: Solve
Example: Solve

4 First order nonlinear differential equations

In this course we only need to know how to solve the separable equations

Example: Solve

Example: Solve

5 Applications

5.1 Mixing problems

Generally, the outflow is well mixed: , where V is the volume of the container.

5.2 Radioactive decay

5.3 1D motion with drag force

where F is the driving force such as gravity, F_d is the drag force.

1. Drag force proportional to velocity: F_d = -kv.
2. Drag force proportional to the square of velocity: if
v > 0.

5.4 1D motion with distance as the independent variable

In 1D motion, if x(t) is a monotonic function (one-to-one), then velocity can be expressed
as a function of distance x. The following formula can be obtained by the chain rule

This transformation from v(t) to v(x) is especially useful when the forcing term is x dependent.
It’s very useful in finding the position where the object comes to rest (v(x) = 0).

6 Euler’s Method

In real applications, most equations we have to solve don’t have analytical solutions. In
this case, we have to resort to numerical methods, in which Euler’s method is the simplest.
Euler’s method is also known as the tangent line method. The basic idea is just linear
approximation. To solve an initial value problem

we use the following recursive procedure

Here h is the step size and Please note that the sequence (y₁, y₂, … ) is only
an approximation to the exact solution (y(t₁), y(t₂), … ).