Editing Linear differential equation (section)

===Second-order case===
A homogeneous linear differential equation of the second order may be written 
<math display="block">y'' + ay' + by = 0,</math>
and its characteristic polynomial is
<math display="block">r^2 + ar + b.</math>

If {{mvar|a}} and {{mvar|b}} are [[real number|real]], there are three cases for the solutions, depending on the discriminant {{math|1=''D'' = ''a''<sup>2</sup> − 4''b''}}. In all three cases, the general solution depends on two arbitrary constants {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}}.

* If {{math|''D'' > 0}}, the characteristic polynomial has two distinct real roots {{mvar|α}}, and {{mvar|β}}. In this case, the general solution is <math display="block">c_1 e^{\alpha x} + c_2 e^{\beta x}.</math>
* If {{math|1=''D'' = 0}}, the characteristic polynomial has a double root {{math|−''a''/2}}, and the general solution is <math display="block">(c_1 + c_2 x) e^{-ax/2}.</math>
* If {{math|''D'' < 0}}, the characteristic polynomial has two [[complex conjugate]] roots {{math|''α'' ± ''βi''}}, and the general solution is <math display="block">c_1 e^{(\alpha + \beta i)x} + c_2 e^{(\alpha - \beta i)x},</math> which may be rewritten in real terms, using [[Euler's formula]] as <math display="block"> e^{\alpha x} (c_1\cos(\beta x) + c_2 \sin(\beta x)).</math>

Finding the solution {{math|''y''(''x'')}} satisfying {{math|1=''y''(0) = ''d''<sub>1</sub>}} and {{math|1=''y''′(0) = ''d''<sub>2</sub>}}, one equates the values of the above general solution at {{math|0}} and its derivative there to {{math|''d''<sub>1</sub>}} and {{math|''d''<sub>2</sub>}}, respectively. This results in a linear system of two linear equations in the two unknowns {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}}. Solving this system gives the solution for a so-called [[Cauchy boundary condition|Cauchy problem]], in which the values at {{math|0}} for the solution of the DEQ and its derivative are specified.