Y2, C7 - Differential Equations

Question 1

How would you separate the variables in: dy/dx = f(x) g(y)

Answer 1

1/g(y) * dy/dx = f(x)
int(1/g(y)) * dy = int(f(x)) * dx

Question 2

Find general solutions to dy/dx = -y/x

Answer 2

1/y * dy/dx = -1/x
int(1/y) * dy = int(-1/x) * dx
lnlyl = -lnlxl + lnlkl
y = k/x

Question 3

Solve d/dx * (ysin(x))

Answer 3

sin * dy/dx + ycos(x)

Question 4

Find general solutions of the equation x^3 dy/dx + 3x^2 * y = sinx

Answer 4

d/dx * (x^3 * y) = sinx
x^3 * y = int(sinx) dx
x^3 * y = -cosx
x^3 * y = -cos(x) + c

Question 5

Using the reverse product rule, find general solutions to 4xy * dy/dx + 2y^2 = x^2

Answer 5

d/dx * (2xy^2) = x^2
2xy^2 = int(x^2)dx
2xy^2 = (1/3) * x^3 + c
y^2 = (1/6) * x^2 + c/2x

Question 6

What is the integrating factor

Answer 6

e^(int(P)dx)

Question 7

How do you solve equations in the form dy/dx + Py = Q
(where P and Q are functions of x)

Answer 7

Reverse product rule

Question 8

What form of equation can you use the reverse product rule on

Answer 8

dy/dx + Py + Q
(where P and Q are functions of x)

Question 9

What is the integrating factor of dy/dx - 4y = e^x

Answer 9

I.F. = e^(int(-4dx))
= e^-4x

Question 10

What do you do once you have found the integrating factor

Answer 10

Multiply all terms in the expression by the integrating factor
Then solve the usual way (reverse product rule)

Question 11

What do you do if there is a coefficient in front of the dy/dx term in an expression of form dy/dx + Py = Q

Answer 11

Divide all terms by the coefficient to remove

Question 12

What would the solution of a * dy/dx + by = 0 be in the form of

Answer 12

y = Ae^(x * -b/a)

Question 13

What would the solution of a(d2y/dx2) + b(dy/dx) + cy = 0 be in the form of

Answer 13

y = Ae^mx

Question 14

What is the Auxiliary equation

Answer 14

am^2 + bm + c = 0

Question 15

What does auxiliary mean

Answer 15

Helpful

Question 16

What is the general solution to a differential equation if the auxiliary equation has two distinct roots

Answer 16

y = Ae^αx + Be^βx

Question 17

What is the auxiliary equation of 2(d2y/dx2) + 5(dy/dx) + 3y = 0

Answer 17

2m^2 + 5m + 3 = 0

Question 18

When the auxiliary equation has two equal roots, what is the general solution

Answer 18

y = (A + Bx)e^αx

Question 19

When the auxiliary equation has no real roots, what is the general equation

Answer 19

y = Ae^αx + Be^βx
(the same as if the roots are real)

Question 20

Find the general solution of d2y/dx2 + 16y = 0

Answer 20

y = Ae^4ix + Be^-4ix
= A(cos4x + isin4x) + B(cos4x - isin4x)
= (A+B)cos4x + (A-B)isin4x
y = Pcos4x + Qsin4x
P = A+B
Q = (A-B)i

Question 21

If the auxiliary equation has two imaginary roots ±Ω, what is the general solution

Answer 21

y = Acos(Ωx) + Bsin(Ωx)
where A and B are arbitrary constants

Question 22

If the auxiliary equation has two complex roots p ±iq, what is the general solution

Answer 22

y = e^px (Acosqx + Bsinqx)
where A and B are arbitrary constants

Question 23

Find the general solution to d2y/dx2 - 6 * dy/dx + 34y = 0

Answer 23

y = Ae^(3+5i)x + Be^(3-5i)x
y = e^3x * (Pcos5x + Qsin5x)

Question 24

What do you do if you have a non-homogeneous second order differential equation = f(x) (not equal to zero on RHS)

Answer 24

Solve as if RHS = 0 to obtain ‘complimentary function’ (C.F.
Then solve equation = f(x), can be found by using appropriate substitutions and comparing coefficients (solution known as particular integral (P.I.))
y = C.F. + P.I. as C.F. = 0 and P.I. = f(x) which sum to f(x)

Question 25

If the form of f(x) is a cos or sin, what should the form of the particular integral be

Answer 25

λcosΩx + μsinΩx

Question 26

What is the particular integral of d2y/dx2 - 5 * dy/dx + 6y = 3

Answer 26

y = λ
dy/dx = 0
d2y/dx2 = 0
6λ = 3
λ = 1/2
P.I. is y = 1/2

Question 27

Find the general solution to d2y/dx2 - 5 * dy/dx + 6y = 3

Answer 27

C.F. = y = Ae^3x + Be^2x
P.I. = 1/2
Therefore:
G.S. = Ae^3x + Be^2x + 1/2

Question 28

What happens if the f(x) is part of the complimentary function e.g. f(x) = e^2x and C.F. = y = Ae^2x + Be^3x

Answer 28

Everything in the P.I. (particular integral) cancels and LHS = 0

Question 29

What should you do if the f(x) is part of your complimentary function

Answer 29

Add an x or sometimes an x^2 in front of your usual P.I. form

Question 30

When should you add an x^2 coefficient to your complimentary function

Answer 30

If there is already an x term and a term with no x coefficient in the auxiliary equation

Question 31

How do you find particular solutions for differential equations when given boundary conditions

Answer 31

Sub in the value and solve simultaneous equations for A and B

Question 32

What should the particular integral be if RHS = ke^px + c

Answer 32

λe^px + μ