Sr Examen
Ecuación diferencial ctgx*cos^2(y)*dx+sin^2(x)*tgy*dy
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Solución
Ha introducido
[src]
2 2 d
cos (y(x))*cot(x) + sin (x)*--(y(x))*tan(y(x)) = 0
dx
$$\sin^{2}{\left(x \right)} \tan{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)} + \cos^{2}{\left(y{\left(x \right)} \right)} \cot{\left(x \right)} = 0$$
sin(x)^2*tan(y)*y' + cos(y)^2*cot(x) = 0
Solución detallada
Tenemos la ecuación:
$$\sin^{2}{\left(x \right)} \tan{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)} + \cos^{2}{\left(y{\left(x \right)} \right)} \cot{\left(x \right)} = 0$$
Esta ecuación diferencial tiene la forma:
donde
$$\operatorname{f_{1}}{\left(x \right)} = \sin^{2}{\left(x \right)}$$
$$\operatorname{g_{1}}{\left(y \right)} = \tan{\left(y{\left(x \right)} \right)}$$
$$\operatorname{f_{2}}{\left(x \right)} = - \cot{\left(x \right)}$$
$$\operatorname{g_{2}}{\left(y \right)} = \cos^{2}{\left(y{\left(x \right)} \right)}$$
Pasemos la ecuación a la forma:
Dividamos ambos miembros de la ecuación en f1(x)
$$\sin^{2}{\left(x \right)}$$
obtendremos
$$\tan{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)} = - \frac{\cos^{2}{\left(y{\left(x \right)} \right)} \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$
Dividamos ambos miembros de la ecuación en g2(y)
$$\cos^{2}{\left(y{\left(x \right)} \right)}$$
obtendremos
$$\frac{\tan{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)}}{\cos^{2}{\left(y{\left(x \right)} \right)}} = - \frac{\cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$
Con esto hemos separado las variables x y y.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$\frac{dx \tan{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)}}{\cos^{2}{\left(y{\left(x \right)} \right)}} = - \frac{dx \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$
o
$$\frac{dy \tan{\left(y{\left(x \right)} \right)}}{\cos^{2}{\left(y{\left(x \right)} \right)}} = - \frac{dx \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$
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.
$$\int \frac{\tan{\left(y \right)}}{\cos^{2}{\left(y \right)}}\, dy = \int \left(- \frac{\cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)\, dx$$
Tomemos estas integrales
$$\frac{1}{2 \cos^{2}{\left(y \right)}} = Const + \frac{1}{2 \sin^{2}{\left(x \right)}}$$
Hemos recibido una ecuación ordinaria con la incógnica y.
(Const - es una constante)
La solución:
$$\operatorname{y_{1}} = y{\left(x \right)} = - \operatorname{acos}{\left(- \sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)} + 2 \pi$$
$$\operatorname{y_{2}} = y{\left(x \right)} = - \operatorname{acos}{\left(\sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)} + 2 \pi$$
$$\operatorname{y_{3}} = y{\left(x \right)} = \operatorname{acos}{\left(- \sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)}$$
$$\operatorname{y_{4}} = y{\left(x \right)} = \operatorname{acos}{\left(\sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)}$$
$$\sin^{2}{\left(x \right)} \tan{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)} + \cos^{2}{\left(y{\left(x \right)} \right)} \cot{\left(x \right)} = 0$$
Esta ecuación diferencial tiene la forma:
f1(x)*g1(y)*y' = f2(x)*g2(y),
donde
$$\operatorname{f_{1}}{\left(x \right)} = \sin^{2}{\left(x \right)}$$
$$\operatorname{g_{1}}{\left(y \right)} = \tan{\left(y{\left(x \right)} \right)}$$
$$\operatorname{f_{2}}{\left(x \right)} = - \cot{\left(x \right)}$$
$$\operatorname{g_{2}}{\left(y \right)} = \cos^{2}{\left(y{\left(x \right)} \right)}$$
Pasemos la ecuación a la forma:
g1(y)/g2(y)*y'= f2(x)/f1(x).
Dividamos ambos miembros de la ecuación en f1(x)
$$\sin^{2}{\left(x \right)}$$
obtendremos
$$\tan{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)} = - \frac{\cos^{2}{\left(y{\left(x \right)} \right)} \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$
Dividamos ambos miembros de la ecuación en g2(y)
$$\cos^{2}{\left(y{\left(x \right)} \right)}$$
obtendremos
$$\frac{\tan{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)}}{\cos^{2}{\left(y{\left(x \right)} \right)}} = - \frac{\cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$
Con esto hemos separado las variables x y y.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$\frac{dx \tan{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)}}{\cos^{2}{\left(y{\left(x \right)} \right)}} = - \frac{dx \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$
o
$$\frac{dy \tan{\left(y{\left(x \right)} \right)}}{\cos^{2}{\left(y{\left(x \right)} \right)}} = - \frac{dx \cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$
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.
$$\int \frac{\tan{\left(y \right)}}{\cos^{2}{\left(y \right)}}\, dy = \int \left(- \frac{\cot{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)\, dx$$
Tomemos estas integrales
$$\frac{1}{2 \cos^{2}{\left(y \right)}} = Const + \frac{1}{2 \sin^{2}{\left(x \right)}}$$
Hemos recibido una ecuación ordinaria con la incógnica y.
(Const - es una constante)
La solución:
$$\operatorname{y_{1}} = y{\left(x \right)} = - \operatorname{acos}{\left(- \sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)} + 2 \pi$$
$$\operatorname{y_{2}} = y{\left(x \right)} = - \operatorname{acos}{\left(\sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)} + 2 \pi$$
$$\operatorname{y_{3}} = y{\left(x \right)} = \operatorname{acos}{\left(- \sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)}$$
$$\operatorname{y_{4}} = y{\left(x \right)} = \operatorname{acos}{\left(\sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)}$$
Respuesta
[src]
/ ________________ \
| / 1 |
y(x) = - acos|- / -------------- *sin(x)| + 2*pi
| / 2 |
\ \/ 1 + C1*sin (x) /
$$y{\left(x \right)} = - \operatorname{acos}{\left(- \sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)} + 2 \pi$$
/ ________________ \
| / 1 |
y(x) = - acos| / -------------- *sin(x)| + 2*pi
| / 2 |
\\/ 1 + C1*sin (x) /
$$y{\left(x \right)} = - \operatorname{acos}{\left(\sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)} + 2 \pi$$
/ ________________ \
| / 1 |
y(x) = acos|- / -------------- *sin(x)|
| / 2 |
\ \/ 1 + C1*sin (x) /
$$y{\left(x \right)} = \operatorname{acos}{\left(- \sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)}$$
/ ________________ \
| / 1 |
y(x) = acos| / -------------- *sin(x)|
| / 2 |
\\/ 1 + C1*sin (x) /
$$y{\left(x \right)} = \operatorname{acos}{\left(\sqrt{\frac{1}{C_{1} \sin^{2}{\left(x \right)} + 1}} \sin{\left(x \right)} \right)}$$
Gráfico para el problema de Cauchy
Clasificación
separable
lie group
separable Integral
Respuesta numérica
[src]
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 3.1415926592729284)
(-5.555555555555555, 6.92262707632273e-310)
(-3.333333333333333, 6.9226280121985e-310)
(-1.1111111111111107, 6.92262707632273e-310)
(1.1111111111111107, 6.92262707632273e-310)
(3.333333333333334, 6.92262707632273e-310)
(5.555555555555557, 6.92262725663693e-310)
(7.777777777777779, 6.9226280310694e-310)
(10.0, 6.9226280310694e-310)
(10.0, 6.9226280310694e-310)