2 - 10 - differential equations

Question 1

first order

Answer 1

• rearrange
• reverse product rule
• IF
• substitution

Question 2

first order - rearrange

Answer 2

get all the y on the side with dy/dx then integrate with respect to x

Question 3

first order - reverses product rule

Answer 3

if its in the form uv’ + u’v = something then uv = something integrated

Question 4

first order - IF

Answer 4

must be in form dy/dx + Py = Q where P and Q are functions of x
multiple by e^ ∫P dx
then will be able to solve

Question 5

first order - substitution

Answer 5

find dy/dx using the substitution
sub in y = z… to the original dy/dx
make the two = and solve

Question 6

homogenous second order differential eq - auxiliary eq

Answer 6

a dy²/dx² + b dy/dx + cy = 0
am² + bm + c = 0

Question 7

homogenous second order- auxiliary eq - b² -4ac >0

Answer 7

2 real roots m1,m2
y = Ae^m1x + Be^m2x

Question 8

homogenous second order- auxiliary eq - b² -4ac =0

Answer 8

1 real repeated root m
y = (A + Bx)e^mx

Question 9

homogenous second order- auxiliary eq - b² -4ac <0

Answer 9

2 imaginary roots
m = p+- iq
y = e^px (Acosqx + Bsinqx)

Question 10

homogenous

Answer 10

every term involves the y term eg a dy²/dx² + b dy/dx + cy = 0

Question 11

non homogenous

Answer 11

not every term involves the y eg dy/dx = y + 2

Question 12

non homogenous second order
a dy²/dx² + b dy/dx + cy = f(x)

Answer 12

• solve a dy²/dx² + b dy/dx + cy = 0 using auxillary eq - this is the complimentary function CF
• solve f(x) - particular integral PI
• general solution y = CF + PI

Question 13

solving f(x) second order eq

Answer 13

A -> PI = λ
Ax + B -> λx + μ
Ax² + Bx + C -> λx² + μx + ⍺
Ae^px -> λe^px
Acos px + Bsin px -> λcos px + μsin px
if f(x) is already part of the CF
eg CF = Ae^2x + B PI y = λ
then times by x PI becomes λx