Differential Equations

Question 1

What are the solutions to these auxiliary equations:
Two distinct roots a, b
Repeated root a
Complex root a+bi

Answer 1

y = Ae^ax + Be^bx
y = (A + Bx)e^ax
y = e^ax (Asinbx + Bcosbx)

Question 2

What is the format of an intergrating factor?

Answer 2

y’ + P(x)y = Q(x)

Question 3

How do you solve an intergrating factor question?

Answer 3

R = e ^ ∫P(x)
Multiply original equation through by R
LHS = d(Ry)/dx

Question 4

What are the trial functions for these functions:
ax+b
ae ^ bx
acosbx / a sinbx

Answer 4

px + q
pe ^ bx
p sinbx + q cosbx
must have both

Question 5

What happens if the trial function is already in the complementary function?

Answer 5

Multiply trial function by x

Question 6

What is the time period for SHM?

Answer 6

T = 2π/ω

Question 7

How do you work out the speed in SHM?

Answer 7

v² = ω²(a² - x²)

Question 8

When is SHM heavily damped?

Answer 8

When auxilary equation has two distinct roots

Question 9

When is SHM lightly damped?

Answer 9

When auxilary equation has two complex roots

Question 10

When is SHM critically damped?

Answer 10

When auxilary equation has repeated roots

Question 11

How do you phrase l’hopitals rule?

Answer 11

Since both f(x)→0 and g(x)→0 l’hopital’s rule can be applied

Question 12

What is l’hopitals rule?

Answer 12

If f(x)/g(x) → ∞/∞ or 0/0 then f(x)/g(x) = f’(x)/g’(x)

Question 13

What is the mean value of a function?

Answer 13

1/(b-a) ∫ f(x) dx between the limits b and a

Question 14

What is the volume of revolution?

Answer 14

π ∫ y² dx

Question 15

What should you do for partial fractions with quadratic denominator?

Answer 15

Bx + C