# second-order linear ODE with constant coefficients

Let’s consider the ordinary second-order linear differential equation

$\frac{{d}^{2}y}{d{x}^{2}}}+a{\displaystyle \frac{dy}{dx}}+by=\mathrm{\hspace{0.33em}0$ | (1) |

which is
homogeneous^{} (http://planetmath.org/HomogeneousLinearDifferentialEquation)
and the coefficients $a,b$ of which are constants. As
mentionned in the entry
“finding another particular solution of linear ODE”, a simple substitution
makes possible to eliminate from it the addend containing first
derivative^{} of the unknown function. Therefore we
concentrate upon the case $a=0$. We have two cases
depending on the sign of $b=\pm {k}^{2}$.

${1}^{\circ}$. $b>0$. We will solve the equation

$\frac{{d}^{2}y}{d{x}^{2}}}+{k}^{2}y=\mathrm{\hspace{0.33em}0}.$ | (2) |

Multiplicating both addends by the expression $2\frac{dy}{dx}$ it becomes

$$2\frac{dy}{dx}\frac{{d}^{2}y}{d{x}^{2}}+2{k}^{2}y\frac{dy}{dx}=\mathrm{\hspace{0.33em}0},$$ |

where the left hand side is the derivative^{} of
${\left(\frac{dy}{dx}\right)}^{2}+{k}^{2}{y}^{2}$. The latter one thus has a constant value
which must be nonnegative; denote it by ${k}^{2}{C}^{2}$. We then have the equation

${\left({\displaystyle \frac{dy}{dx}}\right)}^{2}={k}^{2}({C}^{2}-{y}^{2}).$ | (3) |

After taking the square root and separating the variables it reads

$$\frac{dy}{\pm \sqrt{{C}^{2}-{y}^{2}}}=kdx.$$ |

Integrating (see the table of integrals) this yields

$$\mathrm{arcsin}\frac{y}{C}=k(x-{x}_{0})$$ |

where ${x}_{0}$ is another constant. Consequently, the general solution of the differential equation (2) may be written

$y=C\mathrm{sin}k(x-{x}_{0})$ | (4) |

in which $C$ and ${x}_{0}$ are arbitrary real constants.

If one denotes $C\mathrm{cos}k{x}_{0}={C}_{1}$ and $-C\mathrm{sin}k{x}_{0}={C}_{2}$, then (4) reads

$y={C}_{1}\mathrm{sin}kx+{C}_{2}\mathrm{cos}kx.$ | (5) |

Here, ${C}_{1}$ and ${C}_{2}$ are arbitrary constants. Because both
$\mathrm{sin}kx$ and $\mathrm{cos}kx$ satisfy the given equation (2) and are
linearly independent^{}, its general solution can be written as (5).

${2}^{\circ}$. $$. An analogical treatment of the equation

$\frac{{d}^{2}y}{d{x}^{2}}}-{k}^{2}y=\mathrm{\hspace{0.33em}0}.$ | (6) |

yields for it the general solution

$y={C}_{1}{e}^{kx}+{C}_{2}{e}^{-kx}$ | (7) |

(note that one can eliminate the square root from the equation $y\pm \sqrt{{y}^{2}+C}={C}^{\prime}{e}^{kx}$ and its “inverted equation” $y\mp \sqrt{{y}^{2}+C}=-\frac{C}{{C}^{\prime}}{e}^{-kx}$). The linear independence of the obvious solutions ${e}^{\pm kx}$ implies also the linear independence of $\mathrm{cosh}kx$ and $\mathrm{sinh}kx$ and thus allows us to give the general solution also in the alternative form

$y={C}_{1}\mathrm{sinh}kx+{C}_{2}\mathrm{cosh}kx.$ | (8) |

Remark. The standard method for solving a homogeneous (http://planetmath.org/HomogeneousLinearDifferentialEquation) ordinary second-order linear differential equation (1) with constant coefficients is to use in it the substitution

$y={e}^{rx}$ | (9) |

where $r$ is a constant; see the entry “second order linear
differential equation with constant coefficients”. This method
is possible to use also for such equations of higher order.

## References

- 1 Ernst Lindelöf: Differentiali- ja integralilasku ja sen sovellutukset III.1. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1935).

Title | second-order linear ODE with constant coefficients |
---|---|

Canonical name | SecondorderLinearODEWithConstantCoefficients |

Date of creation | 2014-03-01 17:02:54 |

Last modified on | 2014-03-01 17:02:54 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 8 |

Author | pahio (2872) |

Entry type | Derivation |

Classification | msc 34A05 |