Differential Equations

Question 1

Two different real roots

Answer 1

y = Ae^px + Be^qx

Question 2

One repeated real root

Answer 2

y = (A + Bx)e^px

Question 3

Two complex roots (bi only)

Answer 3

y = Acosbx + Bsinbx

Question 4

Two complex roots (a + bi)

Answer 4

y = e^ax (Acosbx + Bsinbx)

Question 5

Form of f(x) = k

Answer 5

P.I = Lambda

Question 6

Form of f(x) = ax + b

Answer 6

P.I = Lambda + Mu(x)

Question 7

Form of f(x) = ax^2 + bx + c

Answer 7

P.I = Lambda + Mu(x) + Vx^2

Question 8

Form of f(x) = ke^px

Answer 8

P.I = Lambda ((e)^px)

Question 9

Form of f(x) = mcoswx or msinwx or mcoswx + nsinwx

Answer 9

P.I = Lambda (coswx) + Mu (sinwx)

Question 10

Form of f(x) = kxe^px

Answer 10

P.I = Lambda (xe^px) + Mu (e^px)

Question 11

If f(x) appears in CF

Answer 11

Multiply P.I by x, x^2, etc, depending on what is in the CF. e.g, f(x) = dx + e, C.F = y = A + Be^2x, P.I, y = x (Mu (x) + Lambda)

Question 12

Form of the equation when acting under SHM:

Answer 12

x = -w^2x

Question 13

Which equation links velocity and displacement in an equation?

Answer 13

𝑣^2=𝑤^2 (𝑎^2−𝑥^2 )

Question 14

What is the equation for a situation when the particle starts at the centre?

Answer 14

𝑥=𝑎𝑠𝑖𝑛(𝑤𝑡)

Question 15

What is the equation for a situation when the particle starts at the edge?

Answer 15

𝑥=𝑎𝑐𝑜𝑠(𝑤𝑡)

Question 16

What is the method to determine the maximum displacement from the origin?

Answer 16

We need to find the amplitude of the graph, so we have to write it in terms of either sine or cosine using Rsin(x + 𝑎) notation and the expanding this using the double angle formulae and comparing terms on both sides and solving for 𝑎. Then the maximum value with be the value of R as the max value of sine anything will be 1.