Differential Equations

Question 1

Define the ‘order’ of a differential equation

Answer 1

The index of the highest derivative in the equation

Question 2

Define the ‘degree’ of a differential equation

Answer 2

The power of the highest order derivative in the equation

Question 3

When is a differential equation linear

Answer 3

y and it’s derivatives occur in the first degree and in separate terms.

d^2y/d^2x + dy/dx + y

Question 4

Solve a type one first order differential equation, dy/dx = f(x)

Answer 4

Direct integration
y = F(x) + c

Question 5

Solve a type two first order differential equation, dy/dx = f(y)

Answer 5

• Transform
  g(y) = 1/f(y)
• Integrate
  G(y) = x + c

Question 6

Solve a type three first order differential equation, F(y) dy/dx = f(x)

Answer 6

Direct integration
G(y) = F(x) + c

Question 7

Solve a type four first order differential equation, dy/dx + P(x)y = Q(x)

Answer 7

• Multiple by integration factor
  e^ ∫P(x) dx
• Recognise product rule
• Integrate

Question 8

Solve a type five first order differential equation, P(x,y) + Q(x,y) dy/dx = 0 where P(x,y) and Q(x,y) are homogeneous and of the same degree

Answer 8

Rewrite on correct form
Substitution
- y = ux
- dy/dx = u + x du/dx

Question 9

How do you work out if a function is homogeneous

Answer 9

g(kx, ky) = k^m g(x ,y)
(all components have the same power of k, equal number of x and y)

Question 10

Solve a type six first order differential equation, dy/dx + P(x)y = Q(x)y^n

Answer 10

• Divide by y^n
• substitute z = y^ - n+1
  1/(1-n) dz/dx + P(x)z = Q(x)
• type 4