Sr Examen
Ecuación diferencial (y')=cos^2(x-y+5)+1
El profesor se sorprenderá mucho al ver tu solución correcta😉
⌨
Solución
Ha introducido
[src]
d 2 --(y(x)) = 1 + cos (5 + x - y(x)) dx
$$\frac{d}{d x} y{\left(x \right)} = \cos^{2}{\left(x - y{\left(x \right)} + 5 \right)} + 1$$
y' = cos(x - y + 5)^2 + 1
Solución detallada
Tenemos la ecuación:
$$- \cos^{2}{\left(x - y{\left(x \right)} + 5 \right)} + \frac{d}{d x} y{\left(x \right)} - 1 = 0$$
Sustituimos
$$u{\left(x \right)} = x - y{\left(x \right)} + 5$$
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
$$- \cos^{2}{\left(u{\left(x \right)} \right)} + \frac{d}{d x} \left(x - u{\left(x \right)} + 5\right) - 1 = 0$$
o
$$- \cos^{2}{\left(u{\left(x \right)} \right)} - \frac{d}{d x} u{\left(x \right)} = 0$$
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.
$$\int \frac{1}{\cos^{2}{\left(u \right)}}\, du = \int \left(-1\right)\, dx$$
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
$$y{\left(x \right)} = x - u{\left(x \right)} + 5$$
$$y1 = y(x) = x - 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} - 2 C_{1} x + x^{2} + 1} - 1}{C_{1} - x} \right)} + 5$$
$$y2 = y(x) = x + 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} - 2 C_{1} x + x^{2} + 1} + 1}{C_{1} - x} \right)} + 5$$
$$- \cos^{2}{\left(x - y{\left(x \right)} + 5 \right)} + \frac{d}{d x} y{\left(x \right)} - 1 = 0$$
Sustituimos
$$u{\left(x \right)} = x - y{\left(x \right)} + 5$$
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
$$- \cos^{2}{\left(u{\left(x \right)} \right)} + \frac{d}{d x} \left(x - u{\left(x \right)} + 5\right) - 1 = 0$$
o
$$- \cos^{2}{\left(u{\left(x \right)} \right)} - \frac{d}{d x} u{\left(x \right)} = 0$$
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.
$$\int \frac{1}{\cos^{2}{\left(u \right)}}\, du = \int \left(-1\right)\, dx$$
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
$$y{\left(x \right)} = x - u{\left(x \right)} + 5$$
$$y1 = y(x) = x - 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} - 2 C_{1} x + x^{2} + 1} - 1}{C_{1} - x} \right)} + 5$$
$$y2 = y(x) = x + 2 \operatorname{atan}{\left(\frac{\sqrt{C_{1}^{2} - 2 C_{1} x + x^{2} + 1} + 1}{C_{1} - x} \right)} + 5$$
Respuesta
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y(x) = C1 + x*\1 + cos (-5 + C1)/ + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - x *cos (-5 + C1)*sin(-5 + C1) + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + O\x /
3 30 6
$$y{\left(x \right)} = x \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) - x^{2} \sin{\left(C_{1} - 5 \right)} \cos^{3}{\left(C_{1} - 5 \right)} + \frac{x^{3} \left(\left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \left(\left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - \sin^{2}{\left(C_{1} - 5 \right)} + \cos^{2}{\left(C_{1} - 5 \right)}\right) - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + \sin^{2}{\left(C_{1} - 5 \right)} - \cos^{2}{\left(C_{1} - 5 \right)}\right)}{3} + \frac{x^{4} \left(\left(3 \sin^{2}{\left(C_{1} - 5 \right)} - 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \left(\left(- 3 \sin^{2}{\left(C_{1} - 5 \right)} + 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 4 \sin^{2}{\left(C_{1} - 5 \right)} - 6 \cos^{2}{\left(C_{1} - 5 \right)}\right) + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - 4 \sin^{2}{\left(C_{1} - 5 \right)} + 6 \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin{\left(C_{1} - 5 \right)} \cos{\left(C_{1} - 5 \right)}}{6} + \frac{x^{5} \left(- 2 \left(- 3 \sin^{2}{\left(C_{1} - 5 \right)} + 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 2 \left(3 \sin^{2}{\left(C_{1} - 5 \right)} - 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - 2 \left(- \cos^{2}{\left(C_{1} - 5 \right)} - 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + 2 \left(- \cos^{2}{\left(C_{1} - 5 \right)} - 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \left(- 2 \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + 2 \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + 3 \sin^{4}{\left(C_{1} - 5 \right)} - 22 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} + 3 \cos^{4}{\left(C_{1} - 5 \right)} - 2 \cos^{2}{\left(C_{1} - 5 \right)}\right) + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \left(2 \left(- 3 \sin^{2}{\left(C_{1} - 5 \right)} + 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - 2 \left(3 \sin^{2}{\left(C_{1} - 5 \right)} - 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \left(- 2 \left(- \cos^{2}{\left(C_{1} - 5 \right)} - 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + 2 \left(- \cos^{2}{\left(C_{1} - 5 \right)} - 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 3 \sin^{4}{\left(C_{1} - 5 \right)} + 22 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} - 3 \cos^{4}{\left(C_{1} - 5 \right)} + 2 \cos^{2}{\left(C_{1} - 5 \right)}\right) - 2 \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + 2 \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - \left(\left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - \sin^{2}{\left(C_{1} - 5 \right)} + \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(\left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - \sin^{2}{\left(C_{1} - 5 \right)} + \cos^{2}{\left(C_{1} - 5 \right)}\right) \cos^{2}{\left(C_{1} - 5 \right)} + 3 \sin^{4}{\left(C_{1} - 5 \right)} - 22 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} + 3 \cos^{4}{\left(C_{1} - 5 \right)} - 2 \cos^{2}{\left(C_{1} - 5 \right)}\right) + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \left(- 4 \left(3 \sin^{2}{\left(C_{1} - 5 \right)} - 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 2 \left(- \cos^{2}{\left(C_{1} - 5 \right)} - 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - 2 \left(- \cos^{2}{\left(C_{1} - 5 \right)} - 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \left(2 \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - 2 \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 3 \sin^{4}{\left(C_{1} - 5 \right)} + 22 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} - 3 \cos^{4}{\left(C_{1} - 5 \right)} + 2 \cos^{2}{\left(C_{1} - 5 \right)}\right) + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \left(- 4 \left(- 3 \sin^{2}{\left(C_{1} - 5 \right)} + 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \left(2 \left(- \cos^{2}{\left(C_{1} - 5 \right)} - 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - 2 \left(- \cos^{2}{\left(C_{1} - 5 \right)} - 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + 3 \sin^{4}{\left(C_{1} - 5 \right)} - 22 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} + 3 \cos^{4}{\left(C_{1} - 5 \right)} - 2 \cos^{2}{\left(C_{1} - 5 \right)}\right) + 2 \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - 2 \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + \left(\left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - \sin^{2}{\left(C_{1} - 5 \right)} + \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} - \left(\left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - \sin^{2}{\left(C_{1} - 5 \right)} + \cos^{2}{\left(C_{1} - 5 \right)}\right) \cos^{2}{\left(C_{1} - 5 \right)} - 3 \sin^{4}{\left(C_{1} - 5 \right)} + 22 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} - 3 \cos^{4}{\left(C_{1} - 5 \right)} + 2 \cos^{2}{\left(C_{1} - 5 \right)}\right) + \left(- \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + \sin^{2}{\left(C_{1} - 5 \right)} - \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} - \left(- \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + \sin^{2}{\left(C_{1} - 5 \right)} - \cos^{2}{\left(C_{1} - 5 \right)}\right) \cos^{2}{\left(C_{1} - 5 \right)} - 2 \left(\left(- 3 \sin^{2}{\left(C_{1} - 5 \right)} + 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 4 \sin^{2}{\left(C_{1} - 5 \right)} - 6 \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 3 \sin^{4}{\left(C_{1} - 5 \right)} - 22 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 2 \sin^{2}{\left(C_{1} - 5 \right)} + 3 \cos^{4}{\left(C_{1} - 5 \right)} - 2 \cos^{2}{\left(C_{1} - 5 \right)}\right) - \left(- \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + \sin^{2}{\left(C_{1} - 5 \right)} - \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(- \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + \sin^{2}{\left(C_{1} - 5 \right)} - \cos^{2}{\left(C_{1} - 5 \right)}\right) \cos^{2}{\left(C_{1} - 5 \right)} + 2 \left(\left(- 3 \sin^{2}{\left(C_{1} - 5 \right)} + 5 \cos^{2}{\left(C_{1} - 5 \right)}\right) \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) - \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \sin^{2}{\left(C_{1} - 5 \right)} + \left(\cos^{2}{\left(C_{1} - 5 \right)} + 1\right) \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} + 4 \sin^{2}{\left(C_{1} - 5 \right)} - 6 \cos^{2}{\left(C_{1} - 5 \right)}\right) \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - 3 \sin^{4}{\left(C_{1} - 5 \right)} + 22 \sin^{2}{\left(C_{1} - 5 \right)} \cos^{2}{\left(C_{1} - 5 \right)} - 2 \sin^{2}{\left(C_{1} - 5 \right)} - 3 \cos^{4}{\left(C_{1} - 5 \right)} + 2 \cos^{2}{\left(C_{1} - 5 \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, 4.526470558854399)
(-5.555555555555555, 7.044569059902551)
(-3.333333333333333, 9.357540835311152)
(-1.1111111111111107, 11.622947666196536)
(1.1111111111111107, 13.870328551772246)
(3.333333333333334, 16.10900173078909)
(5.555555555555557, 18.342815299112306)
(7.777777777777779, 20.57364330611794)
(10.0, 22.802506688103648)
(10.0, 22.802506688103648)