diplomsko delo
Polona Šenkinc (Author), Marko Slapar (Mentor)

Abstract

Homogene linearne diferencialne enačbe drugega reda so enačbe oblike P(x) y^''+Q(x) y^'+R(x)y=0, kjer je x neodvisna spremenljivka. Takih enačb v splošnem ne znamo reševati. Rešiti znamo le take s konstantnimi koeficienti. Homogene linearne diferencialne enačbe drugega reda, ki imajo za koeficiente predvsem analitične funkcije, pa lahko rešujemo s pomočjo potenčnih vrst. Na začetku je tako navedenih nekaj lastnosti potenčnih vrst, ki jih uporabimo kasneje pri reševanju. Glede na vrednost funkcije P(x) ločimo dve vrsti točk, okoli katerih rešujemo diferencialne enačbe, navadne in singularne točke. Primer enačbe s singularnimi točkami je Eulerjeva enačba x^2 y^''+αxy^'+βy=0, kjer sta α in β realni konstanti. Na primeru Eulerjeve enačbe vidimo, da lahko rešitev zapišemo v določeni obliki, glede na vrednosti ničel karakteristične enačbe F(r)=r(r-1)+αr+β=0. Tako ločimo primere, ko sta ničli realni in različni, realni in enaki ali pa sta konjugiran kompleksni par. Na koncu si ogledamo še Besslovo enačbo x^2 y^''+xy^'+(x^2-ν^2 )y=0, kjer je ν konstanta in njene rešitve reda nič.

Keywords

homogene linearne diferencialne enačbe drugega reda;potenčna vrsta;navadna točka;singularna točka;Eulerjeva enačba;Besslova enačba;

Data

Language: Slovenian
Year of publishing:
Typology: 2.11 - Undergraduate Thesis
Organization: UL PEF - Faculty of Education
Publisher: [P. Šenkinc]
UDC: 51(043.2)
COBISS: 11213897 Link will open in a new window
Views: 878
Downloads: 176
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Other data

Secondary language: English
Secondary title: Solving linear differential equations of second order using power series
Secondary abstract: Second-order linear homogeneous differential equations are mathematical equations of form P(x) y^''+Q(x) y^'+R(x)y=0 where x is an independent variable. In general, we can solve only equations with constant coefficients, and therefore cannot solve the equations in question. On the other hand, second-order linear homogeneous differential equations with coefficients in form of analytic functions can be solved with power series. In dissertation we discuss power series characteristics that we use for solving the equations in question. We can find a series solutions around two types of points, ordinary and singular points. Euler’s equation x^2 y^''+αxy^'+βy=0 is one of the examples where the equation has a singular point and α and β as real constants. When analysing Euler’s equation we find out that the form of the equations solution depends on zero value of characteristic equation (r)=r(r-1)+αr+β=0. There can be different or equal real zeroes or conjugated complex couple of zeroes. In the end, we analyse Bessel’s equation x^2 y^''+xy^'+(x^2-ν^2 )y=0 where ν is a constant with solutions of zero order.
Secondary keywords: mathematics;matematika;
File type: application/pdf
Type (COBISS): Bachelor thesis/paper
Thesis comment: Univ. v Ljubljani, Pedagoška fak., Dvopredmetni učitelj: Fizika-matematika
Pages: 28 str.
ID: 9175291