jungle diff equations

Question 1

solve:
(d/dx)^n y = f(x)

Answer 1

• integrate n times to find y(x)
  (direct integration)

Question 2

solve:
dy/dx = f(x) / g(y)

Answer 2

• move all x’s and y’s to opposite sides then integrate
  (separable equation)

Question 3

solve:
M(x, y)dx + N(x, y)dy = 0

Answer 3

• might be total differential of f(x, y) = c
• if true:
  – N(x, y) = df/dx
  – M(x, y) = df/dy
• check if df/dxdy = df/dydx to confirm
• undo partial derivatives N and M:
  – f(x, y) = int N dx + u(y)
  – f(x, y) = int M dy + v(x)
  – find u(y) or v(x) to get full f(x, y)
  (exact equation)

Question 4

solve:
dy/dx = f(y/x)

Answer 4

• substitute u(x) = y(x)/x
• now dy/dx = u + x du/dx
• should now be separable, you’re welcome :)
  (fake homogeneous equation)

Question 5

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

Answer 5

• multiply by ‘integrating factor’ f(x) = exp(int P(x) dx)
• LHS is now a derivative of products:
  – d/dx [f(x)y] = Q(x)f(x)
• integrate to find y
  (first order linear / integrating factor equation)