Second Order Linear Homogeneous Differential Equations with Constant Coefficients

Consider a differential equation of type

where p, q are some constant coefficients.

For each of the equation we can write the so-called characteristic (auxiliary) equation:

The general solution of the homogeneous differential equation depends on the roots of the characteristic quadratic equation. There are the following options:

1. Discriminant of the characteristic quadratic equation \(D \gt 0.\) Then the roots of the characteristic equations \({k_1}\) and \({k_2}\) are real and distinct. In this case the general solution is given by the following function
   \[y\left( x \right) = {C_1}{e^{{k_1}x}} + {C_2}{e^{{k_2}x}},\]
   where \({C_1}\) and \({C_2}\) are arbitrary real numbers.
2. Discriminant of the characteristic quadratic equation \(D = 0.\) Then the roots are real and equal. It is said in this case that there exists one repeated root \({k_1}\) of order 2. The general solution of the differential equation has the form:
   \[y\left( x \right) = \left( {{C_1}x + {C_2}} \right){e^{{k_1}x}}.\]
3. Discriminant of the characteristic quadratic equation \(D \lt 0.\) Such an equation has complex roots \({k_1} = \alpha + \beta i,\) \({k_2} = \alpha - \beta i.\) The general solution is written as
   \[y\left( x \right) = {e^{\alpha x}}\left[ {{C_1}\cos \left( {\beta x} \right) + {C_2}\sin \left( {\beta x} \right)} \right].\]

Solved Problems

Solution.

First we write the corresponding characteristic equation for the given differential equation:

Eliminate the constant \(C\) from the system of equations:

where \({C_1}\) and \({C_2}\) are arbitrary constants.

Solution.

We write the characteristic equation and calculate its roots:

As it can be seen, the characteristic equation has one root of order \(2:\) \({k_1} = 3.\) Therefore, the general solution of the differential equation is given by

where \({C_1},\) \({C_2}\) are arbitrary real numbers.