How to solve second order linear differential equations that equal zero

How to solve second order linear differential equations that equal zero

Solving Second Order Linear Differential Equations

Basic Theory and Definitions

  • Differential equations deal with functions and their derivatives.
  • A second order differential equation is an equation involving a function and its first and second derivatives.
  • Homogeneous, or zero-order, differential equations have no free term, i.e., they are equal to zero.
  • In a second order linear differential equation of the form ay’’ + by’ + cy = 0, ‘a’, ‘b’, and ‘c’ are constants, and ‘y’ is the function to be found.

Auxiliary Equation

  • The auxiliary equation (also known as the characteristic equation) of the second order linear differential equation ay’’ + by’ + cy = 0 is ar^2 + br + c = 0, where ‘r’ are the roots of the equation, and help solving the differential equation.
  • Finding roots of the auxiliary equation is crucial to solve a differential equation. The form of the solution depends on the nature of these roots.

Case: Distinct Real Roots

  • If the auxiliary equation has two distinct real roots, say r1 and r2, the general solution is y = Ae^(r1x) + Be^(r2x) where A and B are arbitrary constants.

Case: Repeated Real Roots

  • If the auxiliary equation has repeated real roots, i.e., one root r, the general solution is y = (Ax+B)e^rx where A and B are arbitrary constants.

Case: Complex Roots

  • If the auxiliary equation has complex roots, say r = a ± bi, the general solution is y = e^(ax)(Acos(bx) + Bsin(bx)) where A and B are arbitrary constants.

The Process of Solving

  • Write down the auxiliary equation and solve it to find ‘r’.
  • Depending on the roots, write the general solution.
  • If the initial conditions are provided, utilize them to find the values for constants ‘A’ and ‘B’.

Final Remarks

  • Remember that practice is crucial for these problems. Work through many examples to gain a better understanding
  • Always verify your solution by substituting it back into the original differential equation. If it makes the equation true, then the solution is correct.

Tips for Exams

  • Familiarize yourself with the format needed for your final answer.
  • Show all of your workings clearly, explaining each step.
  • Remember to define all constants and functions used.
  • Don’t forget to check your answer with the original question.