diffeq.mws

Diferencialni rovnice 1. radu

Obycejna diferencialni rovnice prvniho radu je rovnice tvaru

y'(x) = f(x,y(x))

pricemz definicni obor f je otevrena podmnozina Rn+1.

Resenim diferencialni rovnice je kazda funkce majici na danem oboru derivaci a splnujici vyse uvedenou rovnost. Protoze kazdym bodem roviny prochazi alespon jedno reseni (blize viz. zde), ma smysl interpretovat reseni dif. rovnice jako vektorove pole.

V nasledujici sekvenci prikladu naleznete vzdy dvojici obrazku k jedne rovnici: prvni z nich ukazuje pole reseni prolozene nekolika konkretnimi funkcemi, druhy je pak animace na niz je reseni vedeno vzdy konkretnim bodem roviny (postupne vsemi body zelene krivky).

1

Zadani:

Diff(y(x),x) = x*y(x)

Reseni (separaci promnenych):

y(x) = _C1*exp(1/2*x^2)

[Maple Plot]

[Maple Plot]

2

Zadani:

Diff(y(x),x) = x/y(x)

Reseni (separaci):

y(x) = sqrt(x^2+_C1), y(x) = -sqrt(x^2+_C1)

[Maple Plot]

[Maple Plot]

3

Zadani:

Diff(y(x),x)-x = 3

Reseni (separaci, f(x)=x+3, g(y)=1):

y(x) = 1/2*x^2+3*x+_C1

[Maple Plot]

[Maple Plot]

4

Zadani:

(1+exp(x))*y(x)*diff(y(x),x) = exp(x)

Reseni (separaci):

y(x) = sqrt(2*ln(1+exp(x))+_C1), y(x) = -sqrt(2*ln(...

[Maple Plot]

[Maple Plot]

5

Zadani:

Diff(y(x),x) = (x+y(x))^2

Reseni (separaci, jde o homogenni rovnici):

y(x) = -tan(-x+_C1)-x

[Maple Plot]

[Maple Plot]

6

Zadani:

Diff(y(x),x) = sin(x)

Reseni (separaci):

y(x) = -cos(x)+_C1

[Maple Plot]

[Maple Plot]

7

Zadani:

Diff(y(x),x)+2*x*y(x) = x^3

Reseni (variaci konstant):

y(x) = 1/2*x^2-1/2+exp(-x^2)*_C1

[Maple Plot]

[Maple Plot]

8

Zadani:

Diff(y(x),x)+x*y(x) = x*y(x)^2

Reseni (Bernoulliova rovnice; variaci konstant):

y(x) = 1/(1+exp(1/2*x^2)*_C1)

[Maple Plot]

[Maple Plot]

9

Zadani:

diff(y(x),x) = x*y(x)^3*sin(x)

Reseni (separaci):

y(x) = sqrt(-(2*cos(x)*sin(x)+2*x*sin(x)^2-2*x-cos(...
y(x) = sqrt(-(2*cos(x)*sin(x)+2*x*sin(x)^2-2*x-cos(...

[Maple Plot]

[Maple Plot]

10

Zadani:

Diff(y(x),x) = 2*x*(x^2+y(x))

Reseni (metodou integracniho faktoru):

y(x) = -x^2-1+exp(x^2)*_C1

[Maple Plot]

[Maple Plot]

11

Zadani:

Diff(y(x),x) = y/x-1

Reseni (homogenni rovnice; separaci):

y(x) = (-ln(x)+_C1)*x

[Maple Plot]

[Maple Plot]