Differential Equations

Question 1

for a differential equation in the format dy/dx + P(x)y = J(x), what is the formula for finding the integrating factor

Answer 1

I.F. = e^int(P(x))

Question 2

what do you do with the integrating factor after you have found it

Answer 2

• you multiply the diff eqn with it

- it should allow you to factorise the product rule

Question 3

what is a linear diff eqn

Answer 3

• an eqn where the variables and derivatives appear as only a linear combination
• it DOESNT matter whether the derivatives are raised to a power

Question 4

what is a homogeneous diff eqn

Answer 4

• an eqn that has no functions of the independent variable appearing on its own (RHS = 0)
• basically if there is a term involving only the independent variable its non-homogeneous

Question 5

is the eqn (dy/dx)^2 + cosy = 15 + x linear/homo

Answer 5

• its not linear because the dependent variable (y) isnt in a linear combination in the cos
• its non-homo because the term ‘x’ only involves the independent variable

Question 6

is the eqn x^3(dy/dx) + 6x^2 + 27 = 0 linear/homo

Answer 6

• its linear because theres no power of y higher than 1

- its non-homo because of the term 6x^2

Question 7

is the eqn (d^3x/dt^3) + (1 + t^2)*(dx/dt) + x = 0 linear/homo

Answer 7

• its linear because theres no power of x higher than 1

- its homo because t is never on its own

Question 8

is the eqn (d^2x/dt^2) + (e^-t/t)*(dx/dt) + cost = 0 linear/homo

Answer 8

• its linear because theres no power of x higher than 1

- its non-homo because of the term cost

Question 9

what is the general formula for the complementory function of a diff eqn with 2 distinct real roots

Answer 9

y = Ae^px + Be^qx

Question 10

what is the general formula for the complementory function of a diff eqn with repeated roots

Answer 10

y = Ae^px + Bxe^px

Question 11

what is the general formula for the complementory function of a diff eqn with complex roots

Answer 11

y = [Ae^px * cos(qx)] + [Be^px * sin(qx)]

Question 12

what kind of damping takes place with real roots, repeated roots and complex roots

Answer 12

• real roots = overdamped
• repeated root = critically damped
• complex roots = underdamped

Question 13

what is the general idea for finding the general solution to a non-homo linear equation

Answer 13

y = complementory function + particular integral

Question 14

if the non-homo eqn has sinx or cosx on the RHS, what would you start guessing with to find the particular integral

Answer 14

y = Asinx + Bcosx

Question 15

for some cases where the RHS = t^2 or something, and just trying x = at^2 doesnt work, what is the next option to try

Answer 15

• a full flown polynomial eqn

- x = at^2 + bt + c

Question 16

if you have no bloody clue where to start with the guess work where do you turn to

Answer 16

the data book and pray somethings in there

Question 17

what do you do when the RHS of the non-homo eqn has the same form as the CF (or one of the term in it)

Answer 17

• you use a similar method when solving homo eqns of repeated roots (petrubed equation)
• basically if your RHS = e^x and your CF has an Ae^x term, you ‘change’ the RHS to e^(1 + e)x
• and examine what happens as e tends to 0
• while going through with the guess in this form (just look at the example)