Sr Examen
Ecuación diferencial cos^2(x)*y*y'=x
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
[src]
2 d
cos (x)*--(y(x))*y(x) = x
dx
$$y{\left(x \right)} \cos^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = x$$
y*cos(x)^2*y' = x
Solución detallada
Tenemos la ecuación:
$$y{\left(x \right)} \cos^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = x$$
Esta ecuación diferencial tiene la forma:
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = \frac{x}{\cos^{2}{\left(x \right)}}$$
$$\operatorname{g_{2}}{\left(y \right)} = \frac{1}{y{\left(x \right)}}$$
Pasemos la ecuación a la forma:
Dividamos ambos miembros de la ecuación en g2(y)
$$\frac{1}{y{\left(x \right)}}$$
obtendremos
$$y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = \frac{x}{\cos^{2}{\left(x \right)}}$$
Con esto hemos separado las variables x y y.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$dx y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = \frac{dx x}{\cos^{2}{\left(x \right)}}$$
o
$$dy y{\left(x \right)} = \frac{dx x}{\cos^{2}{\left(x \right)}}$$
Tomemos la integral de las dos partes de la ecuación:
- de la parte izquierda la integral por y,
- de la parte derecha la integral por x.
$$\int y\, dy = \int \frac{x}{\cos^{2}{\left(x \right)}}\, dx$$
Tomemos estas integrales
$$\frac{y^{2}}{2} = Const - \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$$
Hemos recibido una ecuación ordinaria con la incógnica y.
(Const - es una constante)
La solución:
$$\operatorname{y_{1}} = y{\left(x \right)} = \sqrt{2} \sqrt{\frac{C_{1} \tan^{2}{\left(\frac{x}{2} \right)} - C_{1} - 2 x \tan{\left(\frac{x}{2} \right)} - \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \log{\left(2 \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(2 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}}$$
$$\operatorname{y_{2}} = y{\left(x \right)} = - \sqrt{2} \sqrt{\frac{C_{1} \tan^{2}{\left(\frac{x}{2} \right)} - C_{1} - 2 x \tan{\left(\frac{x}{2} \right)} - \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \log{\left(2 \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(2 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}}$$
$$y{\left(x \right)} \cos^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = x$$
Esta ecuación diferencial tiene la forma:
f1(x)*g1(y)*y' = f2(x)*g2(y),
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = \frac{x}{\cos^{2}{\left(x \right)}}$$
$$\operatorname{g_{2}}{\left(y \right)} = \frac{1}{y{\left(x \right)}}$$
Pasemos la ecuación a la forma:
g1(y)/g2(y)*y'= f2(x)/f1(x).
Dividamos ambos miembros de la ecuación en g2(y)
$$\frac{1}{y{\left(x \right)}}$$
obtendremos
$$y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = \frac{x}{\cos^{2}{\left(x \right)}}$$
Con esto hemos separado las variables x y y.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$dx y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = \frac{dx x}{\cos^{2}{\left(x \right)}}$$
o
$$dy y{\left(x \right)} = \frac{dx x}{\cos^{2}{\left(x \right)}}$$
Tomemos la integral de las dos partes de la ecuación:
- de la parte izquierda la integral por y,
- de la parte derecha la integral por x.
$$\int y\, dy = \int \frac{x}{\cos^{2}{\left(x \right)}}\, dx$$
Tomemos estas integrales
$$\frac{y^{2}}{2} = Const - \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$$
Hemos recibido una ecuación ordinaria con la incógnica y.
(Const - es una constante)
La solución:
$$\operatorname{y_{1}} = y{\left(x \right)} = \sqrt{2} \sqrt{\frac{C_{1} \tan^{2}{\left(\frac{x}{2} \right)} - C_{1} - 2 x \tan{\left(\frac{x}{2} \right)} - \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \log{\left(2 \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(2 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}}$$
$$\operatorname{y_{2}} = y{\left(x \right)} = - \sqrt{2} \sqrt{\frac{C_{1} \tan^{2}{\left(\frac{x}{2} \right)} - C_{1} - 2 x \tan{\left(\frac{x}{2} \right)} - \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \log{\left(2 \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(2 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}}$$
Respuesta
[src]
_______________________________________________________________________________________________________________________________________________________________________________________________
/ / /x\\ / /x\\ 2/x\ 2/x\ / /x\\ 2/x\ / /x\\ 2/x\ 2/x\ / 1 \ /x\ / 1 \
/ -C1 - log|1 + tan|-|| - log|-1 + tan|-|| + C1*tan |-| + tan |-|*log|1 + tan|-|| + tan |-|*log|-1 + tan|-|| - tan |-|*log(2) - tan |-|*log|----------| - 2*x*tan|-| + log(2) + log|----------|
___ / \ \2// \ \2// \2/ \2/ \ \2// \2/ \ \2// \2/ \2/ \1 + cos(x)/ \2/ \1 + cos(x)/
y(x) = \/ 2 * / ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 2/x\
/ -1 + tan |-|
\/ \2/
$$y{\left(x \right)} = \sqrt{2} \sqrt{\frac{C_{1} \tan^{2}{\left(\frac{x}{2} \right)} - C_{1} - 2 x \tan{\left(\frac{x}{2} \right)} - \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \log{\left(2 \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(2 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}}$$
_______________________________________________________________________________________________________________________________________________________________________________________________
/ / /x\\ / /x\\ 2/x\ 2/x\ / /x\\ 2/x\ / /x\\ 2/x\ 2/x\ / 1 \ /x\ / 1 \
/ -C1 - log|1 + tan|-|| - log|-1 + tan|-|| + C1*tan |-| + tan |-|*log|1 + tan|-|| + tan |-|*log|-1 + tan|-|| - tan |-|*log(2) - tan |-|*log|----------| - 2*x*tan|-| + log(2) + log|----------|
___ / \ \2// \ \2// \2/ \2/ \ \2// \2/ \ \2// \2/ \2/ \1 + cos(x)/ \2/ \1 + cos(x)/
y(x) = -\/ 2 * / ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 2/x\
/ -1 + tan |-|
\/ \2/
$$y{\left(x \right)} = - \sqrt{2} \sqrt{\frac{C_{1} \tan^{2}{\left(\frac{x}{2} \right)} - C_{1} - 2 x \tan{\left(\frac{x}{2} \right)} - \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{1}{\cos{\left(x \right)} + 1} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} - \log{\left(2 \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(2 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}}$$
Gráfico para el problema de Cauchy
Clasificación
separable
1st exact
Bernoulli
1st power series
lie group
separable Integral
1st exact Integral
Bernoulli Integral
Respuesta numérica
[src]
(x, y):
(-10.0, 0.75)
(-7.777777777777778, -2.180638301977948e-09)
(-5.555555555555555, 2.17e-322)
(-3.333333333333333, nan)
(-1.1111111111111107, 2.78363573e-315)
(1.1111111111111107, 6.971028255580836e+173)
(3.333333333333334, 3.1933833808213398e-248)
(5.555555555555557, 6.29567287026948e-66)
(7.777777777777779, 8.388243567354547e+296)
(10.0, 3.861029683e-315)
(10.0, 3.861029683e-315)