Differential Equations

Problem

  1. Consider a general linear equation:

dy

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

  • i) Write it in the form M (x, y)dx+N (x, y)dy by taking Q(x) to the LHS and then multiplying the ODE by dx.
  • ii) Use the above form to check when it will be exact.
  • iii) In general, find an integrating factor μ to make this ODE exact.

Problem 2. Solve the following IVP:

(3x2y + 2xy + y3)dx + (x2 + y2)dy = 0, y(0) = 1

Problem 3. Solve the following IVP:

6y′′ − 5y′ + y = 0, y(0) = 1, y′(0) = 1

Problem 4. Solve the following IVP:

y′′ + 4y′ + 4y = 0, y(0) = 1, y′(0) = 3

Problem 5. Find the general solution to the following ODEs:

  1. i) y′′ + y = 0
  2. ii) 2y′′ + 2y′ + y = 0

1Problem 6. Consider the ODE:

y′′′ − y′′ − y′ + y = 0

  • i) Write the characteristic equation associated to this ODE and find the solution(s) to that equation.
  • ii) Analogous to the 2nd order situation, try to find 3 distinct non-zero solutions to the ODE and check that all three are solutions.
  • iii) Guess what the general solution should be