Sr Examen
Ecuación diferencial y'=(sen(x-y+1))^2
El profesor se sorprenderá mucho al ver tu solución correcta😉
⌨
Solución
Ha introducido
[src]
d 2 --(y(x)) = sin (1 + x - y(x)) dx
$$\frac{d}{d x} y{\left(x \right)} = \sin^{2}{\left(x - y{\left(x \right)} + 1 \right)}$$
y' = sin(x - y + 1)^2
Solución detallada
Tenemos la ecuación:
$$- \sin^{2}{\left(x - y{\left(x \right)} + 1 \right)} + \frac{d}{d x} y{\left(x \right)} = 0$$
Sustituimos
$$u{\left(x \right)} = x - y{\left(x \right)} + 1$$
y porque
$$1 - \frac{d}{d x} y{\left(x \right)} = \frac{d}{d x} u{\left(x \right)}$$
entonces
$$\frac{d}{d x} y{\left(x \right)} = 1 - \frac{d}{d x} u{\left(x \right)}$$
sustituimos
$$- \sin^{2}{\left(u{\left(x \right)} \right)} + \frac{d}{d x} \left(x - u{\left(x \right)} + 1\right) = 0$$
o
$$- \sin^{2}{\left(u{\left(x \right)} \right)} - \frac{d}{d x} u{\left(x \right)} + 1 = 0$$
Esta ecuación diferencial tiene la forma:
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(u \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = 1$$
$$\operatorname{g_{2}}{\left(u \right)} = \cos^{2}{\left(u{\left(x \right)} \right)}$$
Pasemos la ecuación a la forma:
Dividamos ambos miembros de la ecuación en g2(u)
$$\cos^{2}{\left(u{\left(x \right)} \right)}$$
obtendremos
$$\frac{\frac{d}{d x} u{\left(x \right)}}{\cos^{2}{\left(u{\left(x \right)} \right)}} = 1$$
Con esto hemos separado las variables x y u.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$\frac{dx \frac{d}{d x} u{\left(x \right)}}{\cos^{2}{\left(u{\left(x \right)} \right)}} = dx$$
o
$$\frac{du}{\cos^{2}{\left(u{\left(x \right)} \right)}} = dx$$
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.
$$\int \frac{1}{\cos^{2}{\left(u \right)}}\, du = \int 1\, dx$$
Tomemos estas integrales
$$\frac{\sin{\left(u \right)}}{\cos{\left(u \right)}} = Const + x$$
Hemos recibido una ecuación ordinaria con la incógnica u.
(Const - es una constante)
La solución:
$$\operatorname{u_{1}} = u{\left(x \right)} = 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} + 2 C_{1} x + x^{2} + 1} - 1}{C_{1} + x} \right)}$$
$$\operatorname{u_{2}} = u{\left(x \right)} = - 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} + 2 C_{1} x + x^{2} + 1} + 1}{C_{1} + x} \right)}$$
hacemos cambio inverso
$$y{\left(x \right)} = x - u{\left(x \right)} + 1$$
$$y1 = y(x) = x - 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} + 2 C_{1} x + x^{2} + 1} - 1}{C_{1} + x} \right)} + 1$$
$$y2 = y(x) = x + 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} + 2 C_{1} x + x^{2} + 1} + 1}{C_{1} + x} \right)} + 1$$
$$- \sin^{2}{\left(x - y{\left(x \right)} + 1 \right)} + \frac{d}{d x} y{\left(x \right)} = 0$$
Sustituimos
$$u{\left(x \right)} = x - y{\left(x \right)} + 1$$
y porque
$$1 - \frac{d}{d x} y{\left(x \right)} = \frac{d}{d x} u{\left(x \right)}$$
entonces
$$\frac{d}{d x} y{\left(x \right)} = 1 - \frac{d}{d x} u{\left(x \right)}$$
sustituimos
$$- \sin^{2}{\left(u{\left(x \right)} \right)} + \frac{d}{d x} \left(x - u{\left(x \right)} + 1\right) = 0$$
o
$$- \sin^{2}{\left(u{\left(x \right)} \right)} - \frac{d}{d x} u{\left(x \right)} + 1 = 0$$
Esta ecuación diferencial tiene la forma:
f1(x)*g1(u)*u' = f2(x)*g2(u),
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(u \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = 1$$
$$\operatorname{g_{2}}{\left(u \right)} = \cos^{2}{\left(u{\left(x \right)} \right)}$$
Pasemos la ecuación a la forma:
g1(u)/g2(u)*u'= f2(x)/f1(x).
Dividamos ambos miembros de la ecuación en g2(u)
$$\cos^{2}{\left(u{\left(x \right)} \right)}$$
obtendremos
$$\frac{\frac{d}{d x} u{\left(x \right)}}{\cos^{2}{\left(u{\left(x \right)} \right)}} = 1$$
Con esto hemos separado las variables x y u.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$\frac{dx \frac{d}{d x} u{\left(x \right)}}{\cos^{2}{\left(u{\left(x \right)} \right)}} = dx$$
o
$$\frac{du}{\cos^{2}{\left(u{\left(x \right)} \right)}} = dx$$
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.
$$\int \frac{1}{\cos^{2}{\left(u \right)}}\, du = \int 1\, dx$$
Tomemos estas integrales
$$\frac{\sin{\left(u \right)}}{\cos{\left(u \right)}} = Const + x$$
Hemos recibido una ecuación ordinaria con la incógnica u.
(Const - es una constante)
La solución:
$$\operatorname{u_{1}} = u{\left(x \right)} = 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} + 2 C_{1} x + x^{2} + 1} - 1}{C_{1} + x} \right)}$$
$$\operatorname{u_{2}} = u{\left(x \right)} = - 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} + 2 C_{1} x + x^{2} + 1} + 1}{C_{1} + x} \right)}$$
hacemos cambio inverso
$$y{\left(x \right)} = x - u{\left(x \right)} + 1$$
$$y1 = y(x) = x - 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} + 2 C_{1} x + x^{2} + 1} - 1}{C_{1} + x} \right)} + 1$$
$$y2 = y(x) = x + 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} + 2 C_{1} x + x^{2} + 1} + 1}{C_{1} + x} \right)} + 1$$
Respuesta
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3 / 2 4 2 2 / 2 2 4 2 2 \ 2 2 \ 5 / 4 4 2 2 2 / 2 2 4 2 2 \ 2 / 2 2 4 4 2 2 \ 2 / 2 2 4 4 2 / 2 4 2 2 2 \ 2 / 4 4 2 2 2 2 \ 2 / 2 4 2 2 2 \ 2 2 2 2 / 2 2 \\ 2 / 2 2 4 4 2 / 2 2 4 2 2 \ 2 / 4 4 2 2 2 2 \ 2 / 4 4 2 2 2 / 2 4 2 2 2 \ 2 / 2 2 4 4 2 2 \ 2 / 2 4 2 2 2 \ 2 2 2 2 / 2 2 \ 2 2 / 2 2 \\ 2 / 2 2 4 2 2 \ 2 2 2 2 / 2 2 \ 2 2 / 2 2 \ 2 2 / 4 2 2 2 / 2 2 \ 2 2 \\ 2 / 2 2 4 2 2 \ 2 2 2 2 / 2 2 \ 2 2 / 4 2 2 2 / 2 2 \ 2 2 \\ 4 / 4 2 2 2 / 2 2 \ 2 / 4 2 2 2 / 2 2 \ 2 2 \ 2 2 \
2 x *\cos (-1 + C1) + sin (-1 + C1) - sin (-1 + C1) + sin (-1 + C1)*\sin (-1 + C1) - cos (-1 + C1) - sin (-1 + C1) + 3*cos (-1 + C1)*sin (-1 + C1)/ - 3*cos (-1 + C1)*sin (-1 + C1)/ x *\- 5*sin (-1 + C1) - 3*cos (-1 + C1) - 2*cos (-1 + C1) + 2*sin (-1 + C1) + cos (-1 + C1)*\sin (-1 + C1) - cos (-1 + C1) - sin (-1 + C1) + 3*cos (-1 + C1)*sin (-1 + C1)/ + sin (-1 + C1)*\- 2*sin (-1 + C1) + 2*cos (-1 + C1) + 3*cos (-1 + C1) + 5*sin (-1 + C1) - 24*cos (-1 + C1)*sin (-1 + C1)/ + sin (-1 + C1)*\- 2*sin (-1 + C1) + 2*cos (-1 + C1) + 3*cos (-1 + C1) + 5*sin (-1 + C1) + cos (-1 + C1)*\cos (-1 + C1) + sin (-1 + C1) - sin (-1 + C1) - 3*cos (-1 + C1)*sin (-1 + C1)/ + sin (-1 + C1)*\- 5*sin (-1 + C1) - 3*cos (-1 + C1) - 2*cos (-1 + C1) + 2*sin (-1 + C1) + 24*cos (-1 + C1)*sin (-1 + C1)/ - sin (-1 + C1)*\cos (-1 + C1) + sin (-1 + C1) - sin (-1 + C1) - 3*cos (-1 + C1)*sin (-1 + C1)/ - 24*cos (-1 + C1)*sin (-1 + C1) - 4*cos (-1 + C1)*sin (-1 + C1)*\2 - 5*sin (-1 + C1) + 3*cos (-1 + C1)// + sin (-1 + C1)*\- 2*sin (-1 + C1) + 2*cos (-1 + C1) + 3*cos (-1 + C1) + 5*sin (-1 + C1) + sin (-1 + C1)*\sin (-1 + C1) - cos (-1 + C1) - sin (-1 + C1) + 3*cos (-1 + C1)*sin (-1 + C1)/ + sin (-1 + C1)*\- 5*sin (-1 + C1) - 3*cos (-1 + C1) - 2*cos (-1 + C1) + 2*sin (-1 + C1) + 24*cos (-1 + C1)*sin (-1 + C1)/ + sin (-1 + C1)*\- 5*sin (-1 + C1) - 3*cos (-1 + C1) - 2*cos (-1 + C1) + 2*sin (-1 + C1) + sin (-1 + C1)*\cos (-1 + C1) + sin (-1 + C1) - sin (-1 + C1) - 3*cos (-1 + C1)*sin (-1 + C1)/ + sin (-1 + C1)*\- 2*sin (-1 + C1) + 2*cos (-1 + C1) + 3*cos (-1 + C1) + 5*sin (-1 + C1) - 24*cos (-1 + C1)*sin (-1 + C1)/ - cos (-1 + C1)*\cos (-1 + C1) + sin (-1 + C1) - sin (-1 + C1) - 3*cos (-1 + C1)*sin (-1 + C1)/ + 24*cos (-1 + C1)*sin (-1 + C1) - 2*cos (-1 + C1)*sin (-1 + C1)*\-2 - 3*cos (-1 + C1) + 5*sin (-1 + C1)/ + 2*cos (-1 + C1)*sin (-1 + C1)*\2 - 5*sin (-1 + C1) + 3*cos (-1 + C1)// - cos (-1 + C1)*\sin (-1 + C1) - cos (-1 + C1) - sin (-1 + C1) + 3*cos (-1 + C1)*sin (-1 + C1)/ - 24*cos (-1 + C1)*sin (-1 + C1) - 2*cos (-1 + C1)*sin (-1 + C1)*\2 - 5*sin (-1 + C1) + 3*cos (-1 + C1)/ + 2*cos (-1 + C1)*sin (-1 + C1)*\-2 - 3*cos (-1 + C1) + 5*sin (-1 + C1)/ + 2*cos (-1 + C1)*sin (-1 + C1)*\-2 - sin (-1 + C1) - 4*cos (-1 + C1) + 6*sin (-1 + C1) + sin (-1 + C1)*\2 - 5*sin (-1 + C1) + 3*cos (-1 + C1)/ + 3*cos (-1 + C1)*sin (-1 + C1)// - sin (-1 + C1)*\sin (-1 + C1) - cos (-1 + C1) - sin (-1 + C1) + 3*cos (-1 + C1)*sin (-1 + C1)/ + 24*cos (-1 + C1)*sin (-1 + C1) - 4*cos (-1 + C1)*sin (-1 + C1)*\-2 - 3*cos (-1 + C1) + 5*sin (-1 + C1)/ - 2*cos (-1 + C1)*sin (-1 + C1)*\-2 - sin (-1 + C1) - 4*cos (-1 + C1) + 6*sin (-1 + C1) + sin (-1 + C1)*\2 - 5*sin (-1 + C1) + 3*cos (-1 + C1)/ + 3*cos (-1 + C1)*sin (-1 + C1)// 2 / 2 \ x *\2 + sin (-1 + C1) - 6*sin (-1 + C1) + 4*cos (-1 + C1) + sin (-1 + C1)*\-2 - 3*cos (-1 + C1) + 5*sin (-1 + C1)/ + sin (-1 + C1)*\-2 - sin (-1 + C1) - 4*cos (-1 + C1) + 6*sin (-1 + C1) + sin (-1 + C1)*\2 - 5*sin (-1 + C1) + 3*cos (-1 + C1)/ + 3*cos (-1 + C1)*sin (-1 + C1)/ - 3*cos (-1 + C1)*sin (-1 + C1)/*cos(-1 + C1)*sin(-1 + C1) / 6\
y(x) = C1 + x*sin (-1 + C1) + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + x *\-1 + sin (-1 + C1)/*cos(-1 + C1)*sin(-1 + C1) + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + O\x /
3 30 6
$$y{\left(x \right)} = x \sin^{2}{\left(C_{1} - 1 \right)} + x^{2} \left(\sin^{2}{\left(C_{1} - 1 \right)} - 1\right) \sin{\left(C_{1} - 1 \right)} \cos{\left(C_{1} - 1 \right)} + \frac{x^{3} \left(\left(- \sin^{4}{\left(C_{1} - 1 \right)} + 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + \sin^{2}{\left(C_{1} - 1 \right)} - \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} + \sin^{4}{\left(C_{1} - 1 \right)} - 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - \sin^{2}{\left(C_{1} - 1 \right)} + \cos^{2}{\left(C_{1} - 1 \right)}\right)}{3} + \frac{x^{4} \left(\left(5 \sin^{2}{\left(C_{1} - 1 \right)} - 3 \cos^{2}{\left(C_{1} - 1 \right)} - 2\right) \sin^{2}{\left(C_{1} - 1 \right)} + \left(\left(- 5 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{2}{\left(C_{1} - 1 \right)} + 2\right) \sin^{2}{\left(C_{1} - 1 \right)} - \sin^{4}{\left(C_{1} - 1 \right)} + 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + 6 \sin^{2}{\left(C_{1} - 1 \right)} - 4 \cos^{2}{\left(C_{1} - 1 \right)} - 2\right) \sin^{2}{\left(C_{1} - 1 \right)} + \sin^{4}{\left(C_{1} - 1 \right)} - 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - 6 \sin^{2}{\left(C_{1} - 1 \right)} + 4 \cos^{2}{\left(C_{1} - 1 \right)} + 2\right) \sin{\left(C_{1} - 1 \right)} \cos{\left(C_{1} - 1 \right)}}{6} + \frac{x^{5} \left(- 4 \left(5 \sin^{2}{\left(C_{1} - 1 \right)} - 3 \cos^{2}{\left(C_{1} - 1 \right)} - 2\right) \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - \left(- \sin^{4}{\left(C_{1} - 1 \right)} + 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + \sin^{2}{\left(C_{1} - 1 \right)} - \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} + \left(- \sin^{4}{\left(C_{1} - 1 \right)} + 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + \sin^{2}{\left(C_{1} - 1 \right)} - \cos^{2}{\left(C_{1} - 1 \right)}\right) \cos^{2}{\left(C_{1} - 1 \right)} + \left(5 \sin^{4}{\left(C_{1} - 1 \right)} - 24 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - 2 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{4}{\left(C_{1} - 1 \right)} + 2 \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} - 2 \left(\left(- 5 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{2}{\left(C_{1} - 1 \right)} + 2\right) \sin^{2}{\left(C_{1} - 1 \right)} - \sin^{4}{\left(C_{1} - 1 \right)} + 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + 6 \sin^{2}{\left(C_{1} - 1 \right)} - 4 \cos^{2}{\left(C_{1} - 1 \right)} - 2\right) \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + \left(- 4 \left(- 5 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{2}{\left(C_{1} - 1 \right)} + 2\right) \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - \left(\sin^{4}{\left(C_{1} - 1 \right)} - 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - \sin^{2}{\left(C_{1} - 1 \right)} + \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} + \left(\sin^{4}{\left(C_{1} - 1 \right)} - 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - \sin^{2}{\left(C_{1} - 1 \right)} + \cos^{2}{\left(C_{1} - 1 \right)}\right) \cos^{2}{\left(C_{1} - 1 \right)} + \left(- 5 \sin^{4}{\left(C_{1} - 1 \right)} + 24 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + 2 \sin^{2}{\left(C_{1} - 1 \right)} - 3 \cos^{4}{\left(C_{1} - 1 \right)} - 2 \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} + 5 \sin^{4}{\left(C_{1} - 1 \right)} - 24 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - 2 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{4}{\left(C_{1} - 1 \right)} + 2 \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} + \left(- 2 \left(- 5 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{2}{\left(C_{1} - 1 \right)} + 2\right) \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + 2 \left(5 \sin^{2}{\left(C_{1} - 1 \right)} - 3 \cos^{2}{\left(C_{1} - 1 \right)} - 2\right) \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + \left(- \sin^{4}{\left(C_{1} - 1 \right)} + 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + \sin^{2}{\left(C_{1} - 1 \right)} - \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} - \left(- \sin^{4}{\left(C_{1} - 1 \right)} + 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + \sin^{2}{\left(C_{1} - 1 \right)} - \cos^{2}{\left(C_{1} - 1 \right)}\right) \cos^{2}{\left(C_{1} - 1 \right)} + \left(- 5 \sin^{4}{\left(C_{1} - 1 \right)} + 24 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + 2 \sin^{2}{\left(C_{1} - 1 \right)} - 3 \cos^{4}{\left(C_{1} - 1 \right)} - 2 \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} + 2 \left(\left(- 5 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{2}{\left(C_{1} - 1 \right)} + 2\right) \sin^{2}{\left(C_{1} - 1 \right)} - \sin^{4}{\left(C_{1} - 1 \right)} + 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + 6 \sin^{2}{\left(C_{1} - 1 \right)} - 4 \cos^{2}{\left(C_{1} - 1 \right)} - 2\right) \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + \left(2 \left(- 5 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{2}{\left(C_{1} - 1 \right)} + 2\right) \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - 2 \left(5 \sin^{2}{\left(C_{1} - 1 \right)} - 3 \cos^{2}{\left(C_{1} - 1 \right)} - 2\right) \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + \left(\sin^{4}{\left(C_{1} - 1 \right)} - 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - \sin^{2}{\left(C_{1} - 1 \right)} + \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} - \left(\sin^{4}{\left(C_{1} - 1 \right)} - 3 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - \sin^{2}{\left(C_{1} - 1 \right)} + \cos^{2}{\left(C_{1} - 1 \right)}\right) \cos^{2}{\left(C_{1} - 1 \right)} + \left(5 \sin^{4}{\left(C_{1} - 1 \right)} - 24 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - 2 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{4}{\left(C_{1} - 1 \right)} + 2 \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} - 5 \sin^{4}{\left(C_{1} - 1 \right)} + 24 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + 2 \sin^{2}{\left(C_{1} - 1 \right)} - 3 \cos^{4}{\left(C_{1} - 1 \right)} - 2 \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} + 5 \sin^{4}{\left(C_{1} - 1 \right)} - 24 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} - 2 \sin^{2}{\left(C_{1} - 1 \right)} + 3 \cos^{4}{\left(C_{1} - 1 \right)} + 2 \cos^{2}{\left(C_{1} - 1 \right)}\right) \sin^{2}{\left(C_{1} - 1 \right)} - 5 \sin^{4}{\left(C_{1} - 1 \right)} + 24 \sin^{2}{\left(C_{1} - 1 \right)} \cos^{2}{\left(C_{1} - 1 \right)} + 2 \sin^{2}{\left(C_{1} - 1 \right)} - 3 \cos^{4}{\left(C_{1} - 1 \right)} - 2 \cos^{2}{\left(C_{1} - 1 \right)}\right)}{30} + C_{1} + O\left(x^{6}\right)$$
Gráfico para el problema de Cauchy
Clasificación
1st power series
lie group
Respuesta numérica
[src]
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 1.563949799844274)
(-5.555555555555555, 3.5372513165619868)
(-3.333333333333333, 5.6773441939929326)
(-1.1111111111111107, 7.8592777888926895)
(1.1111111111111107, 10.057644379120113)
(3.333333333333334, 12.264109627824313)
(5.555555555555557, 14.475153015298217)
(7.777777777777779, 16.68903428175309)
(10.0, 18.904795356908505)
(10.0, 18.904795356908505)