Differential Equations
Define a HOMOGENEOUS differential equation
Every term includes the dependent variable Here is an example
Which direction should you set x to be positive in spring-mass oscillator?
Make the positive x be as you pull the spring away from the equilibrium position so extending the spring (making it longer)
Define a linear differential equation
When each term only includes the dependent variable to the power of 1 or not at all so no y^2 or sin(y) or y*dy/dx
What is the natural length of a spring?
When no forces are acting on the spring
What is d^2x/dt^2 equal to as a product?
d^2x/dt^2 = acceleration so dv/dt d^2x/dt^2 = dv/dt using chain rule dv/dt = dv/dx x dx/dt but dx/dt = velocity so dv/dt = dv/dx x v Therefore: d^2x/dt^2 = dv/dt = dv/dx x v
How would you use the integrating factor method? (easy method without multiplying by anything) e.g cos(x)dy/dx - ysin(x) = x^2
1) Check that equation is linear 2) See if you can spot the reverse product rule in the first two terms it will usually be undifferentiated variable in first term with the undifferentiated variable in the second term e.g d/dx ( ycos(x) ) = x^2 3) integrate both sides with respect to the independent variable e.g ∫d/dx ( ycos(x) ) dx = ∫ x^2 dx y cos(x) = x^3/3 + c so y = (x^3/3 + c) / cos(x)
Set up the differential equation for a horizontal spring-mass operator where the tension in the spring is proportional to the extension
1) To start every question you write F = ma and resolve in the positive x direction so set the spring up in a way that x is positive so the spring has been pulled 2) Sub in values for force, mass and acceleration Force: The only force acting on the spring is tension Tension acts in the opposite direction to the extension so is negative The question tells us that tension is proportional to extension so t = -kx Acceleration: Remember is the second derivative of displacement (x) so is d^2x/dt^2 Acceleration is in the same direction as x so leave it positive Mass: We are not told the mass so just write it as a constant m (if given the mass use that value) 3) Stick it all together -kx = m X d^2x/dt^2 This is a second order, homogenous, linear differential equation
How would you do a spring questions SHM with 2 springs? HELP
1) Turn the paper round each way to make It look vertical and find extensions by doing T1 = T2 - label x, x(0) and y in the same way as you would with a vertical one
Write out a differential equation for the following... The volume, V, of a sphere raindrop is or radius r is decreasing at a rate proportional to its surface area. Find an expression for dr/dt
1) Write out all the derivative you know dV/dt = -k(4πr^2) dV/dr = 4πr^2 2) Use chain rule to find dr/dt dr/dt = dr/dV x dV/dt dr/dt = 1/4πr^2 x -k(4πr^2) 3) Simplify dr/dt = -k
When oscillating where is the velocity the highest?
At the central position (as tension is zero here so zero "resistance")
When oscillating when is the velocity = 0?
At the extremities as the velocity is about to change from positive to negative or vice versa.
How would you solve a forced harmonic motion question?
Do F = ma remembering this force should have an opposite sign to the tension and damping You will need to do the particular integral in this type of question
In the case of dy/dx which is the dependent variable and which is the independent variable
In this case y is the dependent variable and x is the independent variable as is measure of how y changes with respect to x Top variable is - dependent Bottom variable - independent
How do you incorporate damping into the differential equation
It is always proportional to velocity so = k x dx/dt Treat it like another force which acts in the same direction as tension it will give you differential equation where the roots if the auxiliary will be 2 real/2 repeated real/complex conjugate pair - not 2 imaginary This means the general solution not be x = Acos ὦt + Bsin ὦt it will either be y = (Acos(Qx) + Bsin(Qx)) e^(px) y = Ae^λ1x + Be^λ2x y = (A + Bt) e^λt
What does a second order, linear, NON-HOMOGENOUS differential equation with constant coefficients look like?
It looks like a homogenous one with a f(independent variable) stuck at the end y^′′+ay^′+by=f(x)
What is damping in SMH?
It reduces the amplitude of the oscillations over time- acts as resistance or drag so is acting in same direction as tension. Drag is always proportional to velocity so = k x dx/dt
What do you do if your P.I is in the same form as your complementary function e.g CF: y = Ae^2x + Be^3x and P.I: y = ae^3x or CF: y = Acos(2x) + Bsin(2x) and P.I y = Pcos(2x) + Qsin(2x)
Multiply entire P.I by the independent variable e.g CF: y = Ae^2x + Be^3x and P.I: y = axe^3x CF: y = Acos(2x) + Bsin(2x) and P.I y = x(Pcos(2x) + Qsin(2x)) If you have imaginary roots in your auxiliary equation so your CF is e.g e^3x(Acos2x + Bsin2x) and your P.I was e^3x or (Pcos(2x) + Qsin(2x) you would not need to multiply by x as these are not the same.
What is the differential equation to model SHM of a pendulum?
Note that θ should be sinθ but the angle θ is so small that you use small angle approximations where sinθ = θ This shows SHM where ὦ is √g/l
Is the natural length the same as the equilibrium?
Only if the spring is acting HORIZONTALLY on a table with a mass at the end If the spring is hanging VERTICALLY with a mass at then end then the equilibrium position is going to be longer than the natural length and hence you have to introduce a new variable
What five things can this solution tell you about the SHM? Rsin( ὦt + α)
R is the amplitude (max distance from equilibrium) ὦ is the angular frequency α just tells you the shift in the sin graph (oscillation) The period is T and T is 2π/ὦ (time at which motion repeats itself) The frequency is 1/T
What two other ways can you write the solution x = Acos ὦt + Bsin ὦt as?
Rsin( ὦt + α) - write in this form when the SHM starts at the equilibrium position; this makes sense from the sin graph starting at 0. Rcos( ὦt - α) - write in this form when the SHM starts at an extremity position; this makes sense from the cos graph starting at 1. To write in either of these form do the usual R formula method where you expand the Rsin( ὦt + α) or Rcos( ὦt - α) and match R and α
Give one method for solving first order differential equations and when can you use it
Separating the variables You can only do it when dy/dx = f(x)g(y) the functions must be multiplied by each other
In spring oscillators what direction does acceleration in the spring always acts?
Set the acceleration to always be in the same direction as x and the signs will sort themselves out.
Define a NON-HOMOGENEOUS differential equation
Some terms are a function of the independent variable - this could be a constant or a function Every non-homogeneous equation has a homogeneous part - in this case it is dy/dx =y so the non-homogeneous part is 2
Derive the formula which shows the relationship between velocity and displacement.
Start with the differential equation showing SHM Write acceraltion as a product of velocity and dv/dx Solve by separation of variables Find c by putting in boundary condition that v = 0 when x = a (amplitude) rearrange to get formula
What is the extension?
The amount by which the spring has increased from the natural length (usually denoted as x/e if equilibrium position is equal to natural length of spring).
How would you solve linear, SECOND order homogenous with constant coefficient?
The auxiliary method
Define the order of a differential equation
The highest derivative in the differential equation This example would be second order
If you can not use separation of variable ie is not in the form dy/dx = f(x)g(y) what can you use instead and what are the conditions for using this method?
The integrating factor method which you can use if the equation is linear and is in the form dy/dx + P(x)y = Q(x) where P and Q are functions of x
What is simple harmonic motion?
The oscillations (cycle of back and forth motions) of a particle through a central position. The extreme points are equidistance from the centre.
What is the equilibrium position?
The point at which everything is at rest and the spring oscillates around this point
In simple harmonic motion what are you measuring the rate of?
The rate of the extension of spring or displacement of particle from the equilibrium over time
If the damping is under-damped what will the roots/ general solution look like? Draw a graph
The roots will be 2 complex conjugative pairs so the discriminant of the auxiliary equation is < 0
If the damping is critically-damped what will the roots/ general solution look like? Draw a graph
The roots will be 2 repeated real so the discriminant of the auxiliary equation is = 0
In the auxiliary equation if b^2 - 4ac < 0 what form will the general solution be in? (hint there are two ways)
The roots will be imaginary or complex (have real and imaginary parts) If the are imaginary e.g ± Qi then the general solution will be y = Acos(Qx) + Bsin(Qx) If the roots are complex e.g P ± Qi then the general solution will be y = (Acos(Qx) + Bsin(Qx)) e^(px)
If the damping is over-damped what will the roots/ general solution look like? Draw a graph
The roots will be real so the discriminant of the auxiliary equation is > 0
In spring oscillators what direction does tension in the spring always acts?
The tension in the spring is always trying to bring the mass back to the natural length so acts in the opposite direction to the extension of the spring.
What is forced harmonic motion and what does it look like mathematically?
This is when an added force is put into the system which would oppose the tension (restoring force) or damping - so would act in direction of increasing x In the differential equation this look like a non-homogenous part so a function of the independent variable and they would tell you the force as a function of the independent variable
What is the difference between verifying and solving a differential equation?
To verify is something is a solution you just need to stick it in the differential equation you do not need to actually solve the differential equation from scratch and see if the answers are the same When putting it in the differential equation you will need to put it in in x = form and dy/dx = and maybe even second derivative form and see if the LHS = RHS
When you have a trigonometric particular integral how should you find the constants A and B in if y= Acos(3x) + Bsin(3x)
When you have found the first and second derivatives of P.I and subbed into the differential equation, you equate coefficients of cos3x and sin3x and you will form two equations which you can solve simultaneously
In the auxiliary equation if b^2 - 4ac = 0 what form will the general solution be in?
You will have repeated roots so you need to multiply one of the constants by x which is your INDEPENDENT variable (this is similar to when you do this in sequences)
In the auxiliary equation if b^2 - 4ac > 0 what form will the general solution be in?
You will have two distinct roots so...
What is the net force on the mass when in the central position (equilibrium position) when you have a horizitonal spring?
it is zero as no tension in the spring is needed to bring it back to equilibrium as it is already at equilibrium.
At the equilibrium position what is the value of the extension from the equilibrium position NOT the natural length in the horizontal spring?
x = 0
What is the general solution for SHM differential equations and how is this derived?
x = Acos ὦt + Bsin ὦt This comes from solving x'' = ὦ^2 (x) as your auxiliary equation would be λ^2 = -ὦ^2 so λ= ± ὦ i so solution is x = Acos ὦt + Bsin ὦt
If you had a model with a function of t in a bracket to the power of n and you are finding an expression for when t is SMALL not when t = 0 what do you do?
you would do the binomial expansion up to x^1
What is the general differential equation for SHM? How can you relate this to the one we just set up?
-kx = m X d^2x/dt^2 rearrange to get... d^2x/dt^2 = (-k/m)x so our ὦ^2 is (-k/m)
Set up the differential equation for a VERITCAL spring-mass operator where the tension in the spring is proportional to the extension
1) DRAW AND LABEL DIAGRAM ALWAYS LIKE THIS: x is extension from natural position, x(0) is the initial extension which forms the equilibrium position y is the extension from equilibrium position which is what we want to measure oscillations from 2) FIND X(0) You need to find x(0) first if they have not told you it already. Do this by balancing forces in equilibrium so t = mg and we know t = kx so kx = mg and at equilibrium position x = x(0) so kx(0) = mg so x(0) = mg/k 3) WRITE AN EXPRESSION FOR X IN TERMS OF Y AND X(0) This will be x = y + x(0) so subbing in value of x(0) gives x = y + mg/k 4) MAKE THE EXTENSION OF Y (DOWNWARDS DIRECTION) POSITIVE This means your acceleration is positive and so is y therefore, your tension is acting in the negative direction 5) DO F = MA AND TURN INTO SHM DIFFERENTIAL EQUATION Resolve in positive y direction and do F= ma in terms of y Remember the forces acting on the spring are tension AND weight (weight will be acting in positive direction) mg - k[x(0) + y] = m x d^2y/dt^2 mg - k[mg/k + y] = m x d^2y/dt^2 mg - kmg/k - ky = m x d^2y/dt^2 mg - mg - ky = m x d^2y/dt^2 so -ky = m x d^2y/dt^2 whis is in the form of SHM
How do you solve linear systems simultaneously?
1) Differentiate one of the systems (ensure you differentiate each term like you would implicitly) (the system you do not differentiate is the one that will be eliminated) 2) Substitute the other system into the one you just differentiate - you may have to do multiple substitutions to ensure you only have one dependent variable in your equations (e.g either all x's or all y's) 3) You should form a second order differential equation from you your substitution so then solve to find particular solution with boundaries 4) To find the solution for the other system sub your solution into the initial system
How do you find the auxiliary equation and solve to find the general solution?
1) Ensure your differential equation looks like this (similar to a quadratic) a (d^2 y)/(dx^2 )+b dy/dx+cy=0 2) Write out the auxiliary equation by subbing (d^2 y)/(dx^2 ) = λ^2 and dy/dx = λ so aλ^2 + bλ + c = 0 3) Solve as a quadratic to give you the values of λ and put into the form to get the general equation (see next card for what form the general equation takes)
Outline the auxiliary method for a homogeneous second order linear differential equation with constant coefficients.
1) Find auxiliary equation and solve to find general solution 2) Put in boundary conditions to find particular solution
Outline the method of separating the variables
1) For dy/dx = f(x)g(y) put the g(y) on the LHS and integrate with respect to x 2) If it is an indefinite integral then don't forget your constant on the RHS integral
How would you use the integrating factor method? (hard method where you can not spot the reverse chain rule immediately and have to multiply by something)
1) If you can not spot the reverse chain rule immediately you need to multiply by the integrating factor 2) To find the integrating factor you must get your equation into the form dy/dx + P(x)y = Q(x) where dy/dx coefficient is 1 3) The integrating factor is e^(∫P(x) dx) - when finding ∫P(x) dx pretend it is a definite integral and do not worry about +c as they will all cancel out anyways 4)Multiply everything by the integrating factor and you will get a form that means you can use reverse chain rule on it 5) Carry out as you would with the normal solving by integration factor
What are the things to remember when trying to form a first order differential equation?
1) It will usually say the rate of something is proportional to something else so write this as a derivative (rate) = k (thing it is proportional to) 2) If the rate is decreasing then make k negative 3) You may need to use the chain rule 4) You may need to use expressions like surface area of a sphere (4πr^2) or volume of sphere (4/3πr^3) to differentiate 5) If it says rate - it is a derivative where t is the independent variable
How would you solve a second order, linear, NON-HOMOGENOUS differential equation with constant coefficients?
1) Solve the homogeneous part as normal with the auxiliary equation- this is called the complementary function e.g y = Ae^2x + Be^4x 2) Then solve the non-homogeneous part which is called the particular integral To find the particular integral P.I look at the form f(independent variable) Decide on your particular integral e.g y = px + q where p,q are constants Find the first and second derivatives of your P.I sub these into the original differential equation and equate coefficients to find the constants e.g you may find that p = 3 and q = 5 so your P.I is y = 3p + 5 3) Put the complementary function and particular integral together so in this example y = Ae^2x + Be^4x + 3p + 5 4) Put in boundary or initial conditions to find particular solution
Verify that v = 20e^-2t + 5 is the particular solution of the differential equation dv/dt = 10 -2v that satisfies the condition when v= 25, t = 0
1) Start with LHS and differentiate the solution 2) Move to RHS and sub in the solution showing that LHS = RHS 3) Show conditions are satisfied by subbing into solution
