Differential equations

Question 1

f(x) dy/dx + f’(x)y can be written as

Answer 1

d/dx(yf(x))

Question 2

When are integrating factors used?

Answer 2

when a differential equation can be written in the form dy/dx + Py = Q

where P and Q are functions of x

Question 3

integrating factor

Answer 3

e^∫P(x) dx

Question 4

dy/dx α y

Answer 4

dy/dx = ky

Question 5

dy/dx α 1/y

Answer 5

dy/dx = k/y

Question 6

When do you use the auxiliary equation method

Answer 6

ad^2y/dx^2 + bdy/dx + cy = 0

Question 7

auxiliary equation method

Answer 7

1. let y = e^mx
2. sub y, dy/dx, d^2y/dx^2 into the differential equation
3. e^mx>0, factorise to give auxiliary equation

Question 8

what does the auxiliary equation give ?

Answer 8

• for homogeneous differential equations the roots of the auxiliary equation in the correct form give the general solution
• for non-homogeneous it gives the complementary function (part of the general solution)

Question 9

Solution from auxiliary equation for two real roots (α and β)

Answer 9

y = Ae^αx + Be^βx

Question 10

Solution from auxiliary equation for one repeated roots (m)

Answer 10

y = (A+ Bx)e^mx

Question 11

Solution from auxiliary equation for pure imaginary roots (+- ni)

Answer 11

y = Acosnx + Bsinnx

Question 12

Solution from auxiliary equation for a complex conjugate pair (p+-qi)

Answer 12

y = e^px(Acosqx + Bsinqx)

Question 13

how to find a particular integral for a constant on the RHS

Answer 13

1. let y = λ
2. sub y, dy/dx and d^2y/dx^2 into the differential equation
3. solve for λ

Question 14

how to find the particlar integral for a linear function of x on the RHS

Answer 14

1. let y = λx + μ
2. sub y, dy/dx and d^2y/dx^2 into the differential equation
3. solve for λ and μ (equate coefficients)

Question 15

how to find the particular integral for a Quadratic function of x on the RHS

Answer 15

1. let y = λx^2 + μx + ν
2. sub y, dy/dx and d^2y/dx^2 into the differential equation
3. solve for λ, μ and ν (equate coefficients)

Question 16

how to find the particular integral for a exponential function of x on the RHS

Answer 16

1. let y = λe^x
2. sub y, dy/dx and d^2y/dx^2 into the differential equation
3. solve for λ

Question 17

how to find the particular integral for a trigonometric function of x on the RHS

Answer 17

1. let y = λsinx + μcosx
2. sub y, dy/dx and d^2y/dx^2 into the differential equation
3. solve for λ and μ

Question 18

how to find the general solution of a non-homogeneous equation

Answer 18

add the complementary function and the particular integrals

Question 19

simple harmonic motion form

Answer 19

d^2x/dt^2 + ω^2x = 0

Question 20

linear damped systems

Answer 20

there is a force opposing the motion proportional to the speed of the object (dx/dt)

Question 21

roots of auxiliary equation from simple harmonic motion

Answer 21

p+qi

Question 22

if p is negative in the root of the auxiliary equation from simple harmonic motion

Answer 22

underdamping

Question 23

if the roots from the auxiliary equation from simple harmonic motion have real negative roots

Answer 23

overdamping

Question 24

overdamping

Answer 24

no oscillations, dies away slowly

Question 25

underdamping

Answer 25

oscillations with decreasing amplitude

Question 26

simultaneous differential equations method 1

Answer 26

1. rearrange a to give y in terms of x and dx/dt 2. differentiate with respect to t to give dy/dt in terms of dx/dt and d^2x/dt^2 3. substitute expressions for y and dy/dt into equation b and rearrange to give a second order differential equation in x

Question 27

simultaneous differential equations method 2

Answer 27

1. differentiate equation a with respect to t to give d^2x/dt^2 in term of dx/dt and dy/dt 2. use equation b to substitute for dy/dt. This gives an equation for d^2x/dt^2 in terms of x, y and dx/dt 3. rearrange the original equation a to give y in terms of x and dx/dt, and substitute this into the new equation. rearrange to give a second order differential equation in x

Question 28

how to get rid of the implicit differentiation when using integrating factor

Answer 28

integrate the other side

Question 29

auxiliary equation of critically damped

Answer 29

repeated roots

Question 30

auxiliary equation of overdamped

Answer 30

b^2>4ac hence real roots (negative)

Question 31

auxiliary equation of underdamped

Answer 31

- p +- qi