Sr Examen
Ecuación diferencial y'=sen(x-y)
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
[src]
d --(y(x)) = sin(x - y(x)) dx
$$\frac{d}{d x} y{\left(x \right)} = \sin{\left(x - y{\left(x \right)} \right)}$$
y' = sin(x - y)
Solución detallada
Tenemos la ecuación:
$$- \sin{\left(x - y{\left(x \right)} \right)} + \frac{d}{d x} y{\left(x \right)} = 0$$
Sustituimos
$$u{\left(x \right)} = - x + y{\left(x \right)}$$
y porque
$$\frac{d}{d x} y{\left(x \right)} - 1 = \frac{d}{d x} u{\left(x \right)}$$
entonces
$$\frac{d}{d x} y{\left(x \right)} = \frac{d}{d x} u{\left(x \right)} + 1$$
sustituimos
$$\sin{\left(u{\left(x \right)} \right)} + \frac{d}{d x} \left(x + u{\left(x \right)}\right) = 0$$
o
$$\sin{\left(u{\left(x \right)} \right)} + \frac{d}{d x} u{\left(x \right)} + 1 = 0$$
Esta ecuación diferencial tiene la forma:
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(u \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = -1$$
$$\operatorname{g_{2}}{\left(u \right)} = \sin{\left(u{\left(x \right)} \right)} + 1$$
Pasemos la ecuación a la forma:
Dividamos ambos miembros de la ecuación en g2(u)
$$\sin{\left(u{\left(x \right)} \right)} + 1$$
obtendremos
$$\frac{\frac{d}{d x} u{\left(x \right)}}{\sin{\left(u{\left(x \right)} \right)} + 1} = -1$$
Con esto hemos separado las variables x y u.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$\frac{dx \frac{d}{d x} u{\left(x \right)}}{\sin{\left(u{\left(x \right)} \right)} + 1} = - dx$$
o
$$\frac{du}{\sin{\left(u{\left(x \right)} \right)} + 1} = - dx$$
Tomemos la integral de las dos partes de la ecuación:
- de la parte izquierda la integral por u,
- de la parte derecha la integral por x.
$$\int \frac{1}{\sin{\left(u \right)} + 1}\, du = \int \left(-1\right)\, dx$$
Tomemos estas integrales
$$- \frac{2}{\tan{\left(\frac{u}{2} \right)} + 1} = Const - x$$
Hemos recibido una ecuación ordinaria con la incógnica u.
(Const - es una constante)
La solución:
$$\operatorname{u_{1}} = u{\left(x \right)} = - 2 \operatorname{atan}{\left(\frac{C_{1} - x + 2}{C_{1} - x} \right)}$$
hacemos cambio inverso
$$y{\left(x \right)} = x + u{\left(x \right)}$$
$$y1 = y(x) = x - 2 \operatorname{atan}{\left(\frac{C_{1} - x + 2}{C_{1} - x} \right)}$$
$$- \sin{\left(x - y{\left(x \right)} \right)} + \frac{d}{d x} y{\left(x \right)} = 0$$
Sustituimos
$$u{\left(x \right)} = - x + y{\left(x \right)}$$
y porque
$$\frac{d}{d x} y{\left(x \right)} - 1 = \frac{d}{d x} u{\left(x \right)}$$
entonces
$$\frac{d}{d x} y{\left(x \right)} = \frac{d}{d x} u{\left(x \right)} + 1$$
sustituimos
$$\sin{\left(u{\left(x \right)} \right)} + \frac{d}{d x} \left(x + u{\left(x \right)}\right) = 0$$
o
$$\sin{\left(u{\left(x \right)} \right)} + \frac{d}{d x} u{\left(x \right)} + 1 = 0$$
Esta ecuación diferencial tiene la forma:
f1(x)*g1(u)*u' = f2(x)*g2(u),
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(u \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = -1$$
$$\operatorname{g_{2}}{\left(u \right)} = \sin{\left(u{\left(x \right)} \right)} + 1$$
Pasemos la ecuación a la forma:
g1(u)/g2(u)*u'= f2(x)/f1(x).
Dividamos ambos miembros de la ecuación en g2(u)
$$\sin{\left(u{\left(x \right)} \right)} + 1$$
obtendremos
$$\frac{\frac{d}{d x} u{\left(x \right)}}{\sin{\left(u{\left(x \right)} \right)} + 1} = -1$$
Con esto hemos separado las variables x y u.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$\frac{dx \frac{d}{d x} u{\left(x \right)}}{\sin{\left(u{\left(x \right)} \right)} + 1} = - dx$$
o
$$\frac{du}{\sin{\left(u{\left(x \right)} \right)} + 1} = - dx$$
Tomemos la integral de las dos partes de la ecuación:
- de la parte izquierda la integral por u,
- de la parte derecha la integral por x.
$$\int \frac{1}{\sin{\left(u \right)} + 1}\, du = \int \left(-1\right)\, dx$$
Tomemos estas integrales
$$- \frac{2}{\tan{\left(\frac{u}{2} \right)} + 1} = Const - x$$
Hemos recibido una ecuación ordinaria con la incógnica u.
(Const - es una constante)
La solución:
$$\operatorname{u_{1}} = u{\left(x \right)} = - 2 \operatorname{atan}{\left(\frac{C_{1} - x + 2}{C_{1} - x} \right)}$$
hacemos cambio inverso
$$y{\left(x \right)} = x + u{\left(x \right)}$$
$$y1 = y(x) = x - 2 \operatorname{atan}{\left(\frac{C_{1} - x + 2}{C_{1} - x} \right)}$$
Respuesta
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3 / 2 2 / 2 2 \ \ 5 / 2 2 2 / 2 2 \ / 2 2 \ / 2 2 2 2 / 2 2 \ / 2 2 \ \ 2 2 / 2 2 \ / 2 2 2 / 2 2 \ / 2 2 \ / 2 2 / 2 2 \ / 2 2 \ 2 \ / 2 2 \ 2 \ \ 2 4 / 2 2 / 2 2 \ \
x *\sin (C1) - cos (C1) - \cos (C1) - sin (C1) - sin(C1)/*sin(C1) + sin(C1)/ x *\-sin(C1) - 4*sin (C1) + 4*cos (C1) + cos (C1)*(1 + 4*sin(C1)) + \cos (C1) - sin (C1) - sin(C1)/*sin(C1) + \-sin(C1) - 4*sin (C1) + 4*cos (C1)/*sin(C1) + \-sin(C1) - 4*sin (C1) + 4*cos (C1) + cos (C1)*(1 + 4*sin(C1)) - cos (C1)*(-1 - 4*sin(C1)) - \sin (C1) - cos (C1) + sin(C1)/*sin(C1) - \- 4*cos (C1) + 4*sin (C1) + sin(C1)/*sin(C1)/*sin(C1) - cos (C1)*(-1 - 4*sin(C1)) - cos (C1)*\-1 + cos (C1) - sin (C1) - 5*sin(C1) - (1 + 4*sin(C1))*sin(C1)/ - \- 4*cos (C1) + 4*sin (C1) + cos (C1)*\-1 + cos (C1) - sin (C1) - 5*sin(C1) - (1 + 4*sin(C1))*sin(C1)/ + \- 4*cos (C1) + 4*sin (C1) + sin(C1)/*sin(C1) + \- 4*cos (C1) + 4*sin (C1) + \sin (C1) - cos (C1) + sin(C1)/*sin(C1) - \-sin(C1) - 4*sin (C1) + 4*cos (C1)/*sin(C1) - 2*cos (C1)*(1 + 4*sin(C1)) + sin(C1)/*sin(C1) - \cos (C1) - sin (C1) - sin(C1)/*sin(C1) + 2*cos (C1)*(-1 - 4*sin(C1)) + sin(C1)/*sin(C1)/ x *(1 + sin(C1))*cos(C1) x *\-1 + cos (C1) - sin (C1) - 5*sin(C1) + (-1 - 4*sin(C1))*sin(C1) - \1 + sin (C1) - cos (C1) + 5*sin(C1) + (1 + 4*sin(C1))*sin(C1)/*sin(C1)/*cos(C1) / 6\
y(x) = C1 - x*sin(C1) + ---------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------ + O\x /
6 120 2 24
$$y{\left(x \right)} = - x \sin{\left(C_{1} \right)} + \frac{x^{2} \left(\sin{\left(C_{1} \right)} + 1\right) \cos{\left(C_{1} \right)}}{2} + \frac{x^{3} \left(- \left(- \sin^{2}{\left(C_{1} \right)} - \sin{\left(C_{1} \right)} + \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} + \sin^{2}{\left(C_{1} \right)} + \sin{\left(C_{1} \right)} - \cos^{2}{\left(C_{1} \right)}\right)}{6} + \frac{x^{4} \left(\left(- 4 \sin{\left(C_{1} \right)} - 1\right) \sin{\left(C_{1} \right)} - \left(\left(4 \sin{\left(C_{1} \right)} + 1\right) \sin{\left(C_{1} \right)} + \sin^{2}{\left(C_{1} \right)} + 5 \sin{\left(C_{1} \right)} - \cos^{2}{\left(C_{1} \right)} + 1\right) \sin{\left(C_{1} \right)} - \sin^{2}{\left(C_{1} \right)} - 5 \sin{\left(C_{1} \right)} + \cos^{2}{\left(C_{1} \right)} - 1\right) \cos{\left(C_{1} \right)}}{24} + \frac{x^{5} \left(- \left(- 4 \sin{\left(C_{1} \right)} - 1\right) \cos^{2}{\left(C_{1} \right)} + \left(4 \sin{\left(C_{1} \right)} + 1\right) \cos^{2}{\left(C_{1} \right)} + \left(- 4 \sin^{2}{\left(C_{1} \right)} - \sin{\left(C_{1} \right)} + 4 \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} + \left(- \sin^{2}{\left(C_{1} \right)} - \sin{\left(C_{1} \right)} + \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} - \left(- \left(4 \sin{\left(C_{1} \right)} + 1\right) \sin{\left(C_{1} \right)} - \sin^{2}{\left(C_{1} \right)} - 5 \sin{\left(C_{1} \right)} + \cos^{2}{\left(C_{1} \right)} - 1\right) \cos^{2}{\left(C_{1} \right)} + \left(- \left(- 4 \sin{\left(C_{1} \right)} - 1\right) \cos^{2}{\left(C_{1} \right)} + \left(4 \sin{\left(C_{1} \right)} + 1\right) \cos^{2}{\left(C_{1} \right)} - \left(\sin^{2}{\left(C_{1} \right)} + \sin{\left(C_{1} \right)} - \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} - \left(4 \sin^{2}{\left(C_{1} \right)} + \sin{\left(C_{1} \right)} - 4 \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} - 4 \sin^{2}{\left(C_{1} \right)} - \sin{\left(C_{1} \right)} + 4 \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} - \left(2 \left(- 4 \sin{\left(C_{1} \right)} - 1\right) \cos^{2}{\left(C_{1} \right)} - \left(- \sin^{2}{\left(C_{1} \right)} - \sin{\left(C_{1} \right)} + \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} + \left(4 \sin^{2}{\left(C_{1} \right)} + \sin{\left(C_{1} \right)} - 4 \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} + \left(- \left(4 \sin{\left(C_{1} \right)} + 1\right) \sin{\left(C_{1} \right)} - \sin^{2}{\left(C_{1} \right)} - 5 \sin{\left(C_{1} \right)} + \cos^{2}{\left(C_{1} \right)} - 1\right) \cos^{2}{\left(C_{1} \right)} + \left(- 2 \left(4 \sin{\left(C_{1} \right)} + 1\right) \cos^{2}{\left(C_{1} \right)} - \left(- 4 \sin^{2}{\left(C_{1} \right)} - \sin{\left(C_{1} \right)} + 4 \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} + \left(\sin^{2}{\left(C_{1} \right)} + \sin{\left(C_{1} \right)} - \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} + 4 \sin^{2}{\left(C_{1} \right)} + \sin{\left(C_{1} \right)} - 4 \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} + 4 \sin^{2}{\left(C_{1} \right)} + \sin{\left(C_{1} \right)} - 4 \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} - 4 \sin^{2}{\left(C_{1} \right)} - \sin{\left(C_{1} \right)} + 4 \cos^{2}{\left(C_{1} \right)}\right)}{120} + C_{1} + O\left(x^{6}\right)$$
Clasificación
1st power series
lie group