Methods In Differential Equations

Question 1

How do you solve a first-order differential equation that’s in the form dy/dx + p(x)y = q(x)?

(q(x) could be 0 or a constant and dy/dx mustn’t have anything in front of it)

Answer 1

• Calculate your integrating factor = e^{∫p(x) dx} ignore the +c
• Multiply both sides of the original equation by your integrating factor to get a new equation
• Rewrite this new equation in the form
  (Integrating Factor) y = ∫ RHS (of new equation)
• Use boundary conditions to find c

Question 2

How do you solve second order homogeneous differential equations that are in the form
a d²y / dx² + b dy/dx + cy = 0 ?

Answer 2

• Form an auxiliary quadratic by replacing dy/dx with m
• Use the solutions of the auxiliary equation to form a general solution to the equation y =
• Use any given boundary conditions to solve for the constants in the general solution to find the particular solution (if the given boundary conditions are derivatives you willl need to derive the general solution or sub into the original equation)

Question 3

What are the three cases for the auxiliary quadratic?

Answer 3

• If it has two real roots (α and β) y=Ae^αx + Be^βx
• If it has one repeated root (α) y=(A+Bx)e^αx
• If it has complex roots (p±qi) y=e^px(Acos(qx) + Bsin(qx))

Question 4

What are the steps for solving second order non-homogeneous differential equations that are in the form a d²y / dx² + b dy/dx + cy = f(x) ?

Answer 4

• Solve the corresponding homogeneous equation (a d²y / dx² + b dy/dx + cy = 0) to find the Complimentary Function (CF)
• Use f(x) to choose your Particular Integral (PI) (if PI is contained in the CF then you must multipl PI by x until it’s no longer contained in CF)
• Derive PI twice and sub into the original equation
• Solve for the unknowns in the PI
• Your general solution is now y = PI + CF
• Use any given boundary conditions to solve for the constants in the general solution to find the particular solution (if the given boundary conditions are derivatives you willl need to derive the general solution or sub into the original equation)

Question 5

What are the forms your particular integral can take?

Answer 5

If:
f(x) = p then PI = a
f(x) = px+q then PI = ax+b
f(x) = px²+qx+r PI = ax²+bx+c
If f(x) = pe^kx PI = ae^kx
If f(x) = pcos(kx) + qsin(kx) PI = acos(kx) + bsin(kx)

Question 6

When modelling with differential equations what are three things that are useful?

Answer 6

v = ds/dt
a = d²s/dt²

Question 7

What is a good way to state the limitation of a model?

Answer 7

See where in the question they have made an assumption that will make their model inaccurate

Question 8

When solving a differential model (paint of changing concentration in and out for example) what is important?

Answer 8

Write in words whats going in and out then turn this into an equation. Normaly using a general equaiton for the volume and an equation for the concentration

Question 9

What is different about a differential equation that is based on SHM?

Answer 9

• dx/dt = Velocity
• d²x / dt² = Acceleration

Question 10

How would you find the maximum displacement of an object in SHM given you know the particular solution for x?

Answer 10

Carry out R-α and the maximum is the value of x when sin or cos = 1 (this is just R)

Question 11

How would you find the time period of an object in SHM given you know the particular solution for x?

Answer 11

period = 2π/n

Where x = acos(nt) + bsin(nt)

Question 12

How do you know which kind of damping is happening in forced harmonic motion?

Answer 12

If the auxiliary equation has
* 2 real roots - heavy damping
* 1 repeated root - critical damping
* 0 real roots - light damping

Question 13

How do you deal with forced harmonic motion questions?

Answer 13

• Draw a graph (one force in the question will have a given direction, the restoring force acts towards O, the resistive force acts in the opposite direction to the velocity, at least one force must act in the direction of the particle which is generally away from O, the acceleration is always drawn acting in the same direction as velocity even if it doesn’t actually)
• Carry out a horizontal F=ma
• Sub in for a and v (as differentials of x)
• Solve as you normally would a second order non-homogeneous differential equation

Note when t is large this motion becomes SHM

Question 14

How do you deal with coupled first-order differential equations? ( dx/dt = ax+by+f(t) and dy/dt = cx+dy+g(t) )

Answer 14

• If you want to find an equation for x
• Start with the original equation dx/dt and rearrange for y
• DIfferentiate this equation implicitly ( dx/dt becomes d²x/dt², x becomes dx/dt) to find dy/dt
• sub dy/dt and y into the other original equation
• Rearrange
• If you want to find an equation for y
• Start with the original equation dy/dt and rearrange for x
• DIfferentiate this equation implicitly ( dy/dt becomes d²y/dt², y becomes dy/dt) to find dy/dt
• sub dx/dt and x into the other original equation
• Rearrange

Question 15

In coupled first-order differential equations if you already have the general solution to one of the original equations (x= or y=) how can you find the general solution for the other variable?

Answer 15

-If you start with a general solution for x

• Differentiate the general solution you have to get dx/dt
• Sub this into the original equation (the one with dx/dt in)
• rearrange for y

-If you start with a general solution for y

• Differentiate the general solution you have to get dy/dt
• Sub this into the original equation (the one with dy/dt in)
• rearrange for x