Things you need know for the final exam
Chapter 1.
• What is an ordinary differential equation (ODE)?
• Initial value, solution to an ODE, direction fields
• order for an ODE
• linear vs. nonlinear
Chapter 2. First Order ODE
• Four types of first order ODE and the corresponding methods
– First order linear ODE
– separable ODE
dy
dx
dy
dx
+ p(x)y = q(y), integrating factor method
= f (x)h(y), separate y and x on different sides, then integrate
dy
= f (y/x), substitute v = y/x, then transformed to a separable ODE
– homogeneous ODE dx
– exact ODE, M (x, y)dx + N (x, y)dy = 0, where ∂y M (x, y) = ∂x N (x, y).
• Theory: determine the existence interval without solving the equation.
• Autonomous ODE and its equilibrium, as well as stability
• Application: water tank dQ
dt = rate in × concentration in − rate out × concentration out, note Q
is the quantity, not concentration here.
• numerical: Euler’s method. For dy
dt = f (t, y) with y(0) = y0 . Euler’s method approximates
yn+1 = yn + hf (tn , yn ), with tn = nh (h is the time step) for n ≥ 0 and y0 = y(0).
Chapter 3. Second Order ODE
• Theory: determine the existence interval without solving the equation; superposition; Wronskian;
homogeneous vs. nonhomogeneous; fundamental solution;
• constant coefficient homogeneous second order ODE ay 00 + by 0 + cy = 0. Characteristic equation
ar2 + br + c = 0.
– Characteristic equation has two different real roots r1 6= r2 , fundamental solutions are er1 t
and er2 t
– Characteristic equation has two complex roots r1,2 = λ ± iµ, fundamental solutions are
eλt cos(µt) and eλt sin(µt)
– Characteristic equation has one repeated root r1,2 = λ, fundamental solutions are eλt and
teλt
• constant coefficient nonhomogeneous second order ODE ay 00 + by 0 + cy = g(t). For g(t) as polynomial, exponential and cosine, sine functions, the method of undetermined coefficients.
– Depending on the g(t) and the roots for characteristic equation, guess the correct form for
the particular solution (noticing the extra factor ts )
• homogeneous second order ODE: y 00 + p(t)y 0 + q(t)y = 0. If one solution y1 (t) is know, find the
second independent one. The method of reduction of order
• nonhomogeneous second order ODE y 00 + p(t)y 0 + q(t)y = g(t). If the fundamental solution sets
{y1 (t), y2 (t)} are known for the homogeneous part, then we have the variation-of-constant formula.
• application problem: Spring-Mass system. Three factors mu00 + γu0 + ku = 0 for free vibrations.
Know how to determine the three factors. The positive direction.
– γ = 0, no damping, periodic solution.
√
– γ > 0, damping.
If γ > 4mk, over-damped, mass will never cross the equilibrium position;
√
if γ √
= 4mk, critically damped, mass will cross the equilibrium position at most once; if
γ < 4mk, damped vibration, mass will cross the equilibrium position infinite many times.
• Forced vibration: mu00 + γu0 + ku = F (t) with F (t) = A cos wt (or A sin wt)
– if γ > 0, transient solution and steady solution
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Things you need know for the final exam
– if γ = 0, when w =
q
k
m,
resonance happens.
Chapter 4. Higher Order differential equations
• Theory: determine the existence interval; Wronskian; superposition; homogeneous vs. nonhomogeneous
• constant coefficient homogeneous equations: characteristic equations, roots, then the fundamental
solution sets
• constant coefficient nonhomogeneous equations: Method of undetermined coefficient methods, the
rules are similar to 2nd order ODE case
Chapter 6. Laplace transform
• Laplace transform, inverse Laplace transform
• piecewise continuous function and step functions, translation
• Delta (Dirac) function
• convolution
• solve an initial value problem by Laplace transform
• partial fractions
Chapter 7. Linear system
• Homogeneous system ẋ = Ax, x(0) = x0 : eigenvalue and eigen-vector of matrix A
– two different real eigenvalues λ1 6= λ2 , and corresponding eigenvector v1 and v2 , find fundamental solution set
– two complex eigenvalues, λ1,2 = λ ± iµ, and corresponding eigenvector v1 and v2 , find
fundamental solution set
– one repeated real root λ1 = λ2 , find fundamental solution set
• Homogeneous system ẋ = Ax, x(0) = x0 : eigenvalue and eigen-vector of matrix A,
– two different real eigenvalues λ1 6= λ2 , determine the stability of the origin (Saddle points,
Nodes)
– two complex eigenvalues, λ1,2 = λ ± iµ, determine the stability of the origin (Spiral points)
– one repeated real root λ1 = λ2 , determine the stability of the origin (improper node)
g1
• Non-homogeneous system ẋ = Ax + g, for vector g =
where g1 and g2 are exponentials,
g2
cosine, sine, polynomials (including constant), we can use the method of undetermined coefficient
method (Note that in our course, we will face simple questions here, i.e. for the linear system,
no additional factor ts will be included for our problems compared to the single equation case
appearing in Chapter 3 and 4). There are other ways to solve a nonhomogeneous system as above,
such as Laplace transform, variation of constant, digitalization, etc.
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