What is the process to determine if an explicit linear equation is a solution to a SOLDE?
1. Find appropriate derivatives of given equation2. Plug into differential equation to determine if the given equation is a solution
What is the quadratic formula?
λ1 (+)λ2 (-)
What are the three outcomes that can occur from the quadratic formula?
0
What is the process to solve for outcome 1 homogeneous SOLDE?
1. Solve for b^2 - 4ac, 0 > 2. Solve for λ1 (+) λ2 (-)3. Plug into y(x) = (c1)e^(λ1x) + (c2)e^(λ2x)4. Take the derivative5. Plug in inifital value of y and y' to solve for particular solution
What is the process to solve for outcome 2 homogeneous SOLDE?
1. Solve for b^2 - 4ac, = 0 2. Solve for λ1 (+) λ2 (-)3. Plug into y(x) = (c1)e^(λ1x) + (c2)xe^(λ2x)4. Take the derivative5. Plug in ininital value of y and y' to solve for particular solution
What is the process to solve for outcome 3 homogeneous SOLDE?
1. Solve for b^2 - 4ac, 0 < 2. Solve for α and β such that α = (-b/2a) and β = [(4ac-b^2)^1/2]/2a3. Agnkowledge λ1 λ2 where λ1 = α + iβ and λ2 = α - iβ (conjugate)4. Plug into y(x) = [(c1)cos(βx)e^(αx)] + [(c2)sin(βx)e^(αx)]4. Take the derivative5. Plug in initial value of y and y' to solve for particular solution
What is i?
0
What are the components of imaginary numbers?
z = a +ibz = a - ib conjugatea is the real portion and b is the imaginary portion of z
What is the absolute value of imaginary number |z|
(a+b)^(1/2)
What is the equation for a second order homogeneous linear combination?
c1y1(x) + c2y2(x) = 0
What is the Wronstein process for determining if the functions y1 and y2 are linearly independent or dependent?
1. Determine the derivatives of y1 and y22. Plug into 2 x 2 matrix and solve for the determinate y1(x)y2'(x) - y2(x)y1'(x)3. Plug in the interval if necessary4. If W(y1,y2) does not equal 0 then linearly independent else linearly dependent
What is the process to solve for a nonhomogeneous SOLDE given one solution yp(x)?
1. Find the general solution (yh) of the associated homogeneous equation by setting it equal to zero2. Plug into Y(x) = yh(x) + yp(x)
According the the principle of superposition, given two solutions to a given differential equation, can solutions with a similar structure to the solutions given be solved for?
Yes, by plugging in the solutions in replacement of the similar portion of the equation
What is the method of unknown variables?
A way to solve for yp(x)
What types of problems can the method of unknown variables be applied to?
constantpolynomialexponentialtrig functions
What is the method to solve for unknown variables?
1. Determine general form2. Find derivatives of general form3. Simplify using the right side of the equation4. Solve for variables (not including nonletter constants)5. Multiply by smallest integer rule6. Solve for yp(x)5. Plug into general solution Y(x)
What is the general form of a second order polynomial?
a + bx + cx^2
What is the general form for an exponential?
ae^x
What is the general form of trig functions sin(x) or cos(x)
asin(x) + bcos(x)
Method of undetermined coefficients table
0
Method of undetermined coefficients table
where is the smallest of the integers in the differential equation such that y'' + y' = 1 would be multiplied by x because of y'
Why should the general form be determined before the particular form when solving using method of undetermined coefficients?
Used to determine the form of the particular solution being sure that the solution does not fit any of the general forms
What is the particular solution form for method of variation?
yp(x) = c1(x)y1(x) + c2(x)y2(x)
What is the system of equations for method of variation?
c1'y1 + c2'y2 = 0c1'y1' + c2'y2' = f(x)
What are the steps to solving a non homogeneous SOLDE using the method of variation of parameters?
1. Solve yh(x) homogeneous 2. Break the general form into y1(x) and y2(x) not including the constants3. Plug into #244. Solve for c1 and c2 using system of equations5. Integrate c1 and c2 6. Plug into #23 particular solution7. Plug into general solution Y(x)8. Solve for initial value problem if applicable
What are the ways to solve for c1' and c2'?
0
What are the other steps to solving a non homogeneous SOLDE using the method of variation of parameters?
1. Solve yh(x) homogeneous 2. Break the general form into y1(x) and y2(x) not including the constants3. Plug into #26 and solve5. Integrate c1 and c2 6. Plug into #23 particular solution7. Plug into general solution Y(x)8. Solve for initial value problem if applicable
Unit conversion guide (step 1)
0
What is the equation for the change in y vs time of a spring with mass m attached to it.
1. y'' + w^2y = 02. y'' + (k/m)y = 03. y(t) = Asin(wt+p)4. y(t) = c1cos(wt) + c2sin(wt)A = (c1^2+c2^2)^(1/2)sin p = c1/Acos p = c2/Aterm-41tan p = c1/c2w = (k/m)^(1/2)mass (m)coefficient of proportionality from Hooke's law (k)amplitude (A)
What are the equations for period of motion, frequency of motion, and circular frequency of motion?
T = 2π/w (1 cycle per second)f = 1/T (cycle per second)w = (k/m)^(1/2) (radians per second)
What is Hooke's Law?
When a spring stretches, the extension of the spring is proportional to the force stretching it, provided the elastic limit of the spring is not exceeded.Opposite to weightwhere x is how much spring is stretched because of the weight
Describe the graph for simple harmonic motion given A = (17/36)^(1/2)p = 1.82w = 8x(0) = 2/3x'(0) = -1/6
x(0) = initial condition stretchedx'(0) = initial velocity
Types of harmonic motion
1. Simple harmonic motion y'' + w^2y = 0, y'' + (k/m)y = 0, y(t) = Asin(wt+p), y(t) = c1cos(wt) + c2sin(wt)2. Damped vibrations with coefficient B: damped, critically damped, underdamped3. Periodic forced vibrations: damped, not damped (resonance, beats)4. Non-periodic forced vibrations requires Leplace tranformations
What are the steps to solve a simple harmonic function?
0. Conversions remembering downward velocity is + because increasing and upward velocity is - because decreasing1. Solve for k using #31 2. Solve for mass using weight/acceleration in proper units3. Plug into #29.24. Solve the homogeneous equation #29.45. Plug-in initial conditions for additional stretch and velocity6. Solve for amplitude #297. Solve for phi #29 and check for sin cos and tan8. Solve for w #299. Plug into #29.310. Solve for the period and frequency
What is the differential equation for unforced damped vibrations?
0
What are the 3 unforced damped vibration cases and their general solutions?
1. Overdamped B>k2. Critically damped3. Underdamped B<kDampening constant (B)Spring constant (k)w = (k/m)^(1/2)Time constant (τ) = 2m/B2λ = B/m
What is forced vibration and what are the two cases of periodic forced vibration?
Additionally external force with (in this classes case) periodic motion1. No damping B = 02. Damping B ≠ 0frequency of system (w0) = (k/m)^(1/2)frequency of force (w)
What is the differential equation for damping forced vibration?
F(t) not F(x)
What is the differential equation for no damping forced vibration?
0
What are the cases of no damping forced vibration?
1. w0 = w phenomenon of resonance2. w0 ≠ 0 phenomenon of beats
What is the general solution of no damping forced vibration in the case of the phenomenon of beats?
0
What is the general solution of no damping forced vibration in the case of the phenomenon of resonance?
0
What are real life applications of resonance?
1. Swinging such that the person pushing the person on the swing has to match up with the movements of the swing in order for a greater amplitude 2. The vibrations of the guitar strings are amplified within the guitars hollowed wooden potion3. Breaking a wine glass by matching the natural frequency of the win glass determined by its size, shape and composition4. Bridge collapse by marching or air matching the natural frequency of the bridge5. Bumping furniture caused by the frequency of the furniture to match with the frequency of the sound6. Microwave by matching the frequency of water and allowing water to vibrate and evaporate which heats the food
Are beats and resonance present in the case of damped forced vibration?
No
What are Leplace transformations?
In the same way that taking the derivative or integral of a function transforms it, the Leplace transformation transforms a function
What are the conditions that must be met in order for the Leplace transformation to take place?
1. f(t) is piecewise continuous on [0, ∞)2. f(t) should be of exponential order such that |f(t)| is ≤ Me^(st)
Laplace transform table
0
What are the two operational properties of Laplace transformations?
1. Translation2. Derivatives of transformations
What are the steps to solve a Laplace translation transformation?
1. Solve for the Laplace of f(t) 2. Combine e^(at) and e^(-st) like e^(-t(s-a))3. Replace s in Laplace of f(t) with (s-a) and solve
What are the steps to solve a Laplace derivatives of transformation?
1. Solve for the Laplace of f(t)2. Use (-1)^n (d^n)/(ds^n)[f(t)] and solve
What are the steps to solve an IVP with Laplace transforms?
1. Take Laplace of each term2. Combine and plug in initial values3. Solve for Y(s)4. Use algebra or partial fractions to simplify5. Take the inverse Laplace and solve for y(t)
What are the two types of step functions and what are their purposes?
1. unit step function - cut off2. delayed function - shifted
What is the purpose of expressing unit step functions in a compact form?
Manipulation
What is the general form of unit step function B?
0
What is the general form of unit step function A?
0
What is the general form of a delayed function and what does the graph look like?
0
Laplace transformation table
0
What is the Laplace transformation formula for a piece wise function?
0
What are the steps to solve an IVP piecewise problem?
1. Write f(t) piecewise in compact form as shown in #54 #552. Take the Laplace transform of both sides3. Isolate Y(s)4. Simplify using algebra and partial fractions5. Take the inverse laplace of the function6. Write in piecewise notation