Physics I / Second-Order Linear ODEs

Practice question · Put in order

Order the steps for solving a second-order linear ODE with constant coefficients.

Compute the discriminant $b^{2}$ − 4c
Find the roots $r_{1}$ , $r_{2}$
Write the characteristic equation $r^{2}$ + br + $c = 0$
Apply initial conditions to find $C_{1}$ and $C_{2}$
Write the general solution based on root type

Hints

The characteristic equation must be formed before its roots can be used.
Initial conditions are applied last, to fix the arbitrary constants.

Show the answer

Write the characteristic equation $r^{2}$ + br + $c = 0$
Compute the discriminant $b^{2}$ − 4c
Find the roots $r_{1}$ , $r_{2}$
Write the general solution based on root type
Apply initial conditions to find $C_{1}$ and $C_{2}$

Why

This systematic approach works for any constant-coefficient 2nd-order ODE.

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