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Review for First Order Differential Equations1 Integration techniques1.1 Integration by substitution
Example:
Example:
1.2 Integration by parts
Example:
1.3 Integration by partial fractions
2 Existence and uniqueness2.1 Linear Equations For initial value problem
if •the coefficients p(t) and g(t) are both continuous on (a,b), and 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 Example: For initial value problem
1,Find
the largest 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 •(t_{0}, y_{0}) is in R, then there is an open t-interval (c, d), contained in (a, b) and containing t_{0}
(i.e., a ≤ c < Note that: 1. Theorem 2.2 doesn’t give the exact numbers for (c, d), 2.
Theorem 2.2 3 First order linear differential equations3.1 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 You should know how to use integrating factor μ(t) = e^{P(t)} to turn the left hand
side Example: Solve
4 First order nonlinear differential equationsIn this course we only need to know how to solve the separable equations
Example: Solve Example: Solve 5 Applications5.1 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. 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
This transformation from v(t) to v(x) is especially useful when the forcing
term is x dependent. 6 Euler’s MethodIn real applications, most equations we have to solve don’t have analytical
solutions. In
we use the following recursive procedure
Here h is the step size and
Please note that the sequence (y_{1}, y_{2}, … ) is only |