Homogeneous linear differential equation with constant coefficients - Revision history

Vipul: /* Examples with real and complex roots without repetitions */

2012-07-07T00:03:14Z

Examples with real and complex roots without repetitions

← Older revision		Revision as of 00:03, 7 July 2012
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	<math>\frac{-1 \pm \sqrt{3}i}{2}</math>		<math>\frac{-1 \pm \sqrt{3}i}{2}</math>

	The basis solutions for this pair are thus <math>e^{-x/2}\cos(\sqrt{3} x/2}</math> and <math>e^{-x/2}\sin(\sqrt{3} x/2)</math>. The general solution is thus:		The basis solutions for this pair are thus <math>e^{-x/2}\cos(\sqrt{3} x/2)</math> and <math>e^{-x/2}\sin(\sqrt{3} x/2)</math>. The general solution is thus:

	<math>y = C_1e^x + C_2e^{-x/2}\cos(\sqrt{3} x/2) + C_3e^{-x/2}\sin(\sqrt{3}x/2)</math>		<math>y = C_1e^x + C_2e^{-x/2}\cos(\sqrt{3} x/2) + C_3e^{-x/2}\sin(\sqrt{3}x/2)</math>

Vipul at 23:55, 6 July 2012

2012-07-06T23:55:33Z

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 Consider the following polynomial:
-<math>t^k + p_{k-1}t^{k-1} + \dots + p_1t +p_0 = 0</math>
+<math>t^k + p_{k-1}t^{k-1} + \dots + p_1t +p_0</math>
 This polynomial is called the ''characteristic polynomial'' of the differential equation. We consider various cases:
@@ Line 26: / Line 26: @@
 | The general case || For a real root <math>\alpha</math> of multiplicity <math>s</math>, the <math>s</math> basis vectors are <math>x^de^{\alpha x}, 0 \le d < s</math>. For a pair of complex conjugates <math>a \pm ib</math> of multiplicity <math>s</math>, the <math>2s</math> basis vectors are <math>x^de^{ax}\cos(bx), 0 \le d < s</math> and <math>x^de^{ax}\sin(bx), 0 \le d < s</math>.
 |}

Vipul at 23:45, 6 July 2012

2012-07-06T23:45:15Z

← Older revision		Revision as of 23:45, 6 July 2012
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	<math>y^{(k)} + p_{k-1}y^{(k-1)} + \dots + p_1y' + p_0y = 0</math>		<math>y^{(k)} + p_{k-1}y^{(k-1)} + \dots + p_1y' + p_0y = 0</math>

	where <math>p_0,p_1,\dots,p_{k-1}</math> are all constants (i.e., real numbers).		where <math>p_0,p_1,\dots,p_{k-1}</math> are all constants (i.e., real numbers). Here <math>y</math> is the dependent variable and <math>x</math> (which does not appear explicitly above) is the independent variable with respect to which the differentiations occur.

	===Solution method===		===Solution method===

Vipul at 23:44, 6 July 2012

2012-07-06T23:44:05Z

← Older revision		Revision as of 23:44, 6 July 2012
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	\| The polynomial splits completely over the complex numbers into distinct linear factors, but some of the roots are not real \|\| For any real root <math>\alpha</math>, use <math>e^{\alpha x}</math> as a basis vector. Non-real roots occur in complex conjugate pairs. For a pair <math>a \pm ib</math>, choose the vectors <math>e^{ax}\cos(bx)</math> and <math>e^{ax}\sin(bx)</math>. Combining, we get a basis of <math>k</math> vectors.		\| The polynomial splits completely over the complex numbers into distinct linear factors, but some of the roots are not real \|\| For any real root <math>\alpha</math>, use <math>e^{\alpha x}</math> as a basis vector. Non-real roots occur in complex conjugate pairs. For a pair <math>a \pm ib</math>, choose the vectors <math>e^{ax}\cos(bx)</math> and <math>e^{ax}\sin(bx)</math>. Combining, we get a basis of <math>k</math> vectors.
	\|-		\|-
	\| The general case \|\| {{~~fillin~~}} ~~-- combine ideas of two preceding generalizations~~.		\| The general case \|\| For a real root <math>\alpha</math> of multiplicity <math>s</math>, the <math>s</math> basis vectors are <math>x^de^{\alpha x}, 0 \le d < s</math>. For a pair of complex conjugates <math>a \pm ib</math> of multiplicity <math>s</math>, the <math>2s</math> basis vectors are <math>x^de^{ax}\cos(bx), 0 \le d < s</math> and <math>x^de^{ax}\sin(bx), 0 \le d < s</math>.
	\|}		\|}

Vipul: Created page with "==Definition== A '''homogeneous linear differential equation with constant coefficients''', which can also be thought of as a linear differential equation that is simulta..."

2012-07-05T22:50:17Z

Created page with "==Definition== A '''homogeneous linear differential equation with constant coefficients''', which can also be thought of as a linear differential equation that is simulta..."

New page

==Definition==

A '''homogeneous linear differential equation with constant coefficients''', which can also be thought of as a [[linear differential equation]] that is simultaneously an [[autonomous differential equation]], is a differential equation of the form:

<math>y^{(k)} + p_{k-1}y^{(k-1)} + \dots + p_1y' + p_0y = 0</math>

where <math>p_0,p_1,\dots,p_{k-1}</math> are all constants (i.e., real numbers).

===Solution method===

Consider the following polynomial:

<math>t^k + p_{k-1}t^{k-1} + \dots + p_1t +p_0 = 0</math>

This polynomial is called the ''characteristic polynomial'' of the differential equation. We consider various cases:

{| class="sortable" border="1"
! Case !! Solution in that case
|-
| The polynomial has pairwise distinct real roots <math>\alpha_1, \alpha_2, \dots, \alpha_k</math> || The solution space has basis <math>e^{\alpha_1 x}, e^{\alpha_2 x}, \dots, e^{\alpha_k x}</math>. In other words, the general solution is <math>C_1e^{\alpha_1x} + C_2e^{\alpha_2x} + \dots + C_ke^{\alpha_kx}</math> where <math>C_1,C_2,\dots,C_k</math> are freely varying real parameters.
|-
| The polynomial splits completely into linear factors over the reals, but with possible repetitions. <math>\alpha_1</math> occurs <math>s_1</math> times, <math>\alpha_2</math> occurs <math>s_2</math> times, and so on till <math>\alpha_r</math>, which occurs <math>s_r</math> times. We have <math>s_1 + s_2 + \dots + s_r = k</math>. || The solution space has basis all functions of the form <math>x^de^{\alpha_ix}</math> where <math>0 \le d < s_i</math> with <math>d</math> an integer. Thus, for each <math>i</math>, there are <math>s_i</math> basis vectors corresponding to <math>\alpha_i</math>. We get a total of <math>k</math> basis vectors.
|-
| The polynomial splits completely over the complex numbers into distinct linear factors, but some of the roots are not real || For any real root <math>\alpha</math>, use <math>e^{\alpha x}</math> as a basis vector. Non-real roots occur in complex conjugate pairs. For a pair <math>a \pm ib</math>, choose the vectors <math>e^{ax}\cos(bx)</math> and <math>e^{ax}\sin(bx)</math>. Combining, we get a basis of <math>k</math> vectors.
|-
| The general case || {{fillin}} -- combine ideas of two preceding generalizations.
|}