Differential Equations

Question 1

Describe the seperation of variables method.

What are some good tips for this?

Answer 1

1. Seperate x and y’s.
2. Integrate.

When seperating, look for things that can be simplified, e.g. common quadratics, common factors etc.

Question 2

What is the general form for a non-seperable first order DE?

How can you solve these equations?

Answer 2

dy/dx + P(x)y = Q(x)

dy/dx must have a coefficient of 1.

You will need to find the integrating factor.

ρ = e^∫P(x)dx

Multiply through by ρ (you will often find the LHS is an exact derivative)

Integrate.

Question 3

What are the 3 possible cases when solving second order DE’s.

What are there solutions?

Answer 3

When finding the roots;

i) (r₁ ≠ r₂) — use y = Ae^r₁x + Be^r₂x
ii) (r₁ = r₂) — use y = (A + Bx)e^rx
iii) Complex roots (r = a + ßi) — use y = e^ax(Acosßx + Bsinßx)

Question 4

What are the 3 possible solutions to second order DEs?

In what situationis each used?

Answer 4

y = Ae^{<strong>r₁</strong>x} + Be^{<strong>r₂</strong>x} (r₁ ≠ r₂)

y = (A + Bx)e^{<strong>r</strong>x} (r₁ = r₂)

y = e^{<strong>a</strong>x}(Acosßx + Bsinßx) (r = a + ßi)