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 fy(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) = eP(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, Fd is the drag force.
1. Drag force proportional to velocity: Fd = -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), … ).
