Homogeneous ODE

Homogeneous ODEs are where the differential equation looks like:

$$a_n(t)x^n+a_{n-1}(t)x^{n-1}+\cdots +a_1(t)x^{\prime }+a_0(t)x=0$$

If the right-hand side of the equation were some function $g(t)$, we would be discussing a non-homogeneous ODE. Homogeneous ODEs are really easy to solve in comparison.

An immediate candidate solution is $x=e^{st}$. Substituting it (and its derivatives) into the equation, we can then factor out $e^{st}$. Take a second-order equation as an example:

$$e^{st}(a{s}^2+bs+c)=0$$

Since $e^{st}$ cannot be 0, the roots are determined by the polynomial:

$$a{s}^2+bs+c=0$$

The polynomial is the characteristic equation (or auxiliary equation). The general solution of the homogeneous equation is generally called the zero-input response or the natural response.

Methods

• Reduction of order, for second-order linear ODEs