Witam,
\(\displaystyle{ N \mbox{d}x +M \mbox{d}y=(x^2y^3+y) \mbox{d}x +(x^3y^2-x) \mbox{d}y=0}\)
\(\displaystyle{ N_y=3y^2x^2+1}\),\(\displaystyle{ M_x=3y^2x^2-1}\)
\(\displaystyle{ N_y-M_x=2 \neq 0}\)
Szukam czynnika całkowego postaci \(\displaystyle{ \mu(\sqrt{xy})}\)
\(\displaystyle{ \mu_x=\frac{\sqrt y}{2\sqrt{x}}\mu'}\),\(\displaystyle{ \mu_y=\frac{\sqrt x}{2\sqrt{y}}\mu'}\)
Teraz wymnazam róœnanie wyjściowe przez czynnik i różniczkuje
\(\displaystyle{ \mu(N_y-M_x)=2\mu=\mu_xM-\mu_yN=\mu' \frac{\sqrt y}{2\sqrt{x}}M-\mu'\frac{\sqrt x}{2\sqrt{y}}N=\break= \mu'\frac{\sqrt y}{2\sqrt{x}}(x^3y^2-x) -\mu'\frac{\sqrt x}{2\sqrt{y}}(x^2y^3+y)=\mu'\frac{1}{2}((xy)^{\frac{5}{2}}-\sqrt{xy})-\mu'\frac{1}{2}((xy)^{\frac{5}{2}}+\sqrt{xy})=\break =-\mu'\sqrt{xy}}\)
Mamy równanie
\(\displaystyle{ 2\mu=-\mu'\sqrt{xy}}\),niech \(\displaystyle{ t=\sqrt{xy}}\). Wtedy rownanie ma postac (\(\displaystyle{ \mu}\) jest funkcja zmiennej \(\displaystyle{ t=\sqrt{xy}}\))
\(\displaystyle{ \frac{ \mbox{d}\mu }{\mu}= \frac{-2 \mbox{d}t }{t}}\) którego rozwiązaniem jest np
\(\displaystyle{ \mu(t)=t^{-2}}\)
Zatem czynnik calkujacy to \(\displaystyle{ \mu(x,y)= \frac{1}{xy}}\)
Ale niestety,
\(\displaystyle{ (\mu N)_y=\left( \frac{x^2y^2+1}{x}\right)_y \neq \left( \mu M\right)_x=\left( \frac{x^3y^2-x}{xy} \right)_x =\left( \frac{x^2y^2-1}{y} \right)_x}\)
co widac od razu
Co poszło nie tak?
Różniczka zupełna czynnikiem całkowym
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janusz47
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Różniczka zupełna czynnikiem całkowym
\(\displaystyle{ M(x,y):= M, \ \ N(x,y) := N, \ \ \mu(x,y) := \mu.}\)
\(\displaystyle{ \mu M dx + \mu N dy = 0}\) (1)
\(\displaystyle{ (\mu M)_{|y} = ( \mu N)_{|x}}\) (2)
\(\displaystyle{ \mu_{y}M + \mu M_{y} = \mu_{x} N - \mu N_{x}}\) (3)
\(\displaystyle{ \mu_{y}M - \mu_{x}N = \mu ( N_{x} - M_{y})}\) (4)
Kładąc
\(\displaystyle{ N_{x} - M_{y} = R(z)( xM - yM); \ \ z = xy}\) (5)
\(\displaystyle{ \mu_{y} M - \mu_{x}N = R(z)(\mu x M - \mu y N)}\) (6)
\(\displaystyle{ \begin{cases} \mu_{y} = \mu x R(z)\\ \mu_{x} = \mu_{y}R(z) \end{cases}}\) (7)
\(\displaystyle{ \mu = \mu(x,y) = \mu(z)}\) (8)
\(\displaystyle{ \mu_{x}(z) = \frac{\partial \mu}{\partial z}\cdot \frac{\partial z}{\partial x}=\mu'(z) y}\) (9)
\(\displaystyle{ \mu_{y}(z) = \frac{\partial \mu}{\partial z}\cdot \frac{ \partial z}{\partial y}= \mu'(z) x}\) (10)
\(\displaystyle{ \begin{cases} \mu'(z) x = \mu(z) x R(z) \\ \mu'(z)y = \mu(z) y R(z) \end{cases}}\) (11)
\(\displaystyle{ \mu'(z) = \mu(z) R(z) \rightarrow \frac{d\mu}{\mu} = R(z)dz}\) (12)
\(\displaystyle{ \mu(z) = e^{\int R(z)dz }}\) (13)
\(\displaystyle{ \frac{N_{x} - M_{y}}{xM - yN} = R(xy) = R(z)}\) (14)
\(\displaystyle{ (x^2 y^3 + y)dx + ( x^3 y^2 - x) dy = 0}\) (15)
Z (14):
\(\displaystyle{ R(x,y) = \frac{3x^2-1 -3x^2 -1}{x^3y^3 +xy -x^3y^3 +xy}= \frac{-2}{2xy}=\frac{-1}{xy}= -\frac{1}{z}}\) (16)
Z ( 13):
\(\displaystyle{ \mu(z) = e^{-\int \frac{1}{z}dz} = e^{-\ln(z)}= \frac{1}{z}= \frac{1}{xy}}\) (17)
\(\displaystyle{ \frac{1}{xy}\left( x^2 y^3 +y \right )dx + \frac{1}{xy}\left( x^3y^2 -x\right) dy= 0}\) (18)
\(\displaystyle{ \left( xy^2 + \frac{1}{x}\right)dx + \left ( x^2y - \frac{1}{y} \right) dy = 0}\) (19)
\(\displaystyle{ \overline{M}_{y}= 2xy = \overline {N}_{x}= 2xy}\)
Proszę znaleźć całkę ogólną równania zupełnego (19).
\(\displaystyle{ \mu M dx + \mu N dy = 0}\) (1)
\(\displaystyle{ (\mu M)_{|y} = ( \mu N)_{|x}}\) (2)
\(\displaystyle{ \mu_{y}M + \mu M_{y} = \mu_{x} N - \mu N_{x}}\) (3)
\(\displaystyle{ \mu_{y}M - \mu_{x}N = \mu ( N_{x} - M_{y})}\) (4)
Kładąc
\(\displaystyle{ N_{x} - M_{y} = R(z)( xM - yM); \ \ z = xy}\) (5)
\(\displaystyle{ \mu_{y} M - \mu_{x}N = R(z)(\mu x M - \mu y N)}\) (6)
\(\displaystyle{ \begin{cases} \mu_{y} = \mu x R(z)\\ \mu_{x} = \mu_{y}R(z) \end{cases}}\) (7)
\(\displaystyle{ \mu = \mu(x,y) = \mu(z)}\) (8)
\(\displaystyle{ \mu_{x}(z) = \frac{\partial \mu}{\partial z}\cdot \frac{\partial z}{\partial x}=\mu'(z) y}\) (9)
\(\displaystyle{ \mu_{y}(z) = \frac{\partial \mu}{\partial z}\cdot \frac{ \partial z}{\partial y}= \mu'(z) x}\) (10)
\(\displaystyle{ \begin{cases} \mu'(z) x = \mu(z) x R(z) \\ \mu'(z)y = \mu(z) y R(z) \end{cases}}\) (11)
\(\displaystyle{ \mu'(z) = \mu(z) R(z) \rightarrow \frac{d\mu}{\mu} = R(z)dz}\) (12)
\(\displaystyle{ \mu(z) = e^{\int R(z)dz }}\) (13)
\(\displaystyle{ \frac{N_{x} - M_{y}}{xM - yN} = R(xy) = R(z)}\) (14)
\(\displaystyle{ (x^2 y^3 + y)dx + ( x^3 y^2 - x) dy = 0}\) (15)
Z (14):
\(\displaystyle{ R(x,y) = \frac{3x^2-1 -3x^2 -1}{x^3y^3 +xy -x^3y^3 +xy}= \frac{-2}{2xy}=\frac{-1}{xy}= -\frac{1}{z}}\) (16)
Z ( 13):
\(\displaystyle{ \mu(z) = e^{-\int \frac{1}{z}dz} = e^{-\ln(z)}= \frac{1}{z}= \frac{1}{xy}}\) (17)
\(\displaystyle{ \frac{1}{xy}\left( x^2 y^3 +y \right )dx + \frac{1}{xy}\left( x^3y^2 -x\right) dy= 0}\) (18)
\(\displaystyle{ \left( xy^2 + \frac{1}{x}\right)dx + \left ( x^2y - \frac{1}{y} \right) dy = 0}\) (19)
\(\displaystyle{ \overline{M}_{y}= 2xy = \overline {N}_{x}= 2xy}\)
Proszę znaleźć całkę ogólną równania zupełnego (19).