Sr Examen
Ecuación diferencial 2*y*y"=y'⁵+7
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Solución
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2 5
d /d \
2*---(y(x))*y(x) = 7 + |--(y(x))|
2 \dx /
dx
$$2 y{\left(x \right)} \frac{d^{2}}{d x^{2}} y{\left(x \right)} = \left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7$$
2*y*y'' = y'^5 + 7
Solución detallada
Tenemos la ecuación:
$$2 y{\left(x \right)} \frac{d^{2}}{d x^{2}} y{\left(x \right)} = \left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7$$
Esta ecuación diferencial tiene la forma:
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y' \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = \frac{1}{2 y{\left(x \right)}}$$
$$\operatorname{g_{2}}{\left(y' \right)} = \left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7$$
Pasemos la ecuación a la forma:
Dividamos ambos miembros de la ecuación en g2(y')
$$\left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7$$
obtendremos
$$\frac{\frac{d^{2}}{d x^{2}} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7} = \frac{1}{2 y{\left(x \right)}}$$
Con esto hemos separado las variables x y y'.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$\frac{dx \frac{d^{2}}{d x^{2}} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7} = \frac{dx}{2 y{\left(x \right)}}$$
o
$$\frac{dy'}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7} = \frac{dx}{2 y{\left(x \right)}}$$
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.
$$\int \frac{1}{y'^{5} + 7}\, dy' = \int \frac{1}{2 y{\left(x \right)}}\, dx$$
Tomemos estas integrales
$$\operatorname{RootSum} {\left(7503125 t^{5} - 1, \left( t \mapsto t \log{\left(35 t + y' \right)} \right)\right)} = Const + \frac{\int \frac{1}{y{\left(x \right)}}\, dx}{2}$$
Hemos recibido una ecuación ordinaria con la incógnica y'.
(Const - es una constante)
La solución:
$$\operatorname{y'1} = - \frac{2 \sqrt[5]{7} \log{\left(\operatorname{y'}{\left(x \right)} + \sqrt[5]{7} \right)}}{35} - 2 \left(- \frac{\sqrt[5]{7}}{140} + \frac{\sqrt{5} \sqrt[5]{7}}{140} + \frac{\left(-1\right) \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{35}\right) \log{\left(\operatorname{y'}{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)} - 2 \left(- \frac{\sqrt[5]{7}}{140} + \frac{\sqrt{5} \sqrt[5]{7}}{140} + \frac{\sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{35}\right) \log{\left(\operatorname{y'}{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)} - 2 \left(- \frac{\sqrt{5} \sqrt[5]{7}}{140} - \frac{\sqrt[5]{7}}{140} + \frac{\left(-1\right) \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{35}\right) \log{\left(\operatorname{y'}{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)} - 2 \left(- \frac{\sqrt{5} \sqrt[5]{7}}{140} - \frac{\sqrt[5]{7}}{140} + \frac{\sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{35}\right) \log{\left(\operatorname{y'}{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)} = C_{1} - \int \frac{1}{y{\left(x \right)}}\, dx$$
tomemos estas integrales
$$\operatorname{y_{1}} = \int \left(- 2 \operatorname{RootSum} {\left(7503125 t^{5} - 1, \left( t \mapsto t \log{\left(35 t + \frac{d}{d x} y{\left(x \right)} \right)} \right)\right)}\right)\, dx = \int \left(C_{1} - \int \frac{1}{y{\left(x \right)}}\, dx\right)\, dx$$ =
$$\operatorname{y_{1}} = - \frac{\sqrt[5]{7} \left(\int \sqrt{5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\, dx + \int \sqrt{5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\, dx + \int \left(- \sqrt{5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx + \int \left(- \sqrt{5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx + \int \left(- \sqrt{2} i \sqrt{5 - \sqrt{5}} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx + \int \sqrt{2} i \sqrt{5 - \sqrt{5}} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\, dx + \int \left(- \sqrt{2} i \sqrt{\sqrt{5} + 5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\right)\, dx + \int \sqrt{2} i \sqrt{\sqrt{5} + 5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\, dx + \int 4 \log{\left(\frac{d}{d x} y{\left(x \right)} + \sqrt[5]{7} \right)}\, dx + \int \left(- \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\right)\, dx + \int \left(- \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\right)\, dx + \int \left(- \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx + \int \left(- \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx\right)}{70} = C_{2} + \int \left(C_{1} - \int \frac{1}{y{\left(x \right)}}\, dx\right)\, dx$$
$$2 y{\left(x \right)} \frac{d^{2}}{d x^{2}} y{\left(x \right)} = \left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7$$
Esta ecuación diferencial tiene la forma:
f1(x)*g1(y')*y'' = f2(x)*g2(y'),
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y' \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = \frac{1}{2 y{\left(x \right)}}$$
$$\operatorname{g_{2}}{\left(y' \right)} = \left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7$$
Pasemos la ecuación a la forma:
g1(y')/g2(y')*y''= f2(x)/f1(x).
Dividamos ambos miembros de la ecuación en g2(y')
$$\left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7$$
obtendremos
$$\frac{\frac{d^{2}}{d x^{2}} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7} = \frac{1}{2 y{\left(x \right)}}$$
Con esto hemos separado las variables x y y'.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$\frac{dx \frac{d^{2}}{d x^{2}} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7} = \frac{dx}{2 y{\left(x \right)}}$$
o
$$\frac{dy'}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{5} + 7} = \frac{dx}{2 y{\left(x \right)}}$$
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.
$$\int \frac{1}{y'^{5} + 7}\, dy' = \int \frac{1}{2 y{\left(x \right)}}\, dx$$
Tomemos estas integrales
$$\operatorname{RootSum} {\left(7503125 t^{5} - 1, \left( t \mapsto t \log{\left(35 t + y' \right)} \right)\right)} = Const + \frac{\int \frac{1}{y{\left(x \right)}}\, dx}{2}$$
Hemos recibido una ecuación ordinaria con la incógnica y'.
(Const - es una constante)
La solución:
$$\operatorname{y'1} = - \frac{2 \sqrt[5]{7} \log{\left(\operatorname{y'}{\left(x \right)} + \sqrt[5]{7} \right)}}{35} - 2 \left(- \frac{\sqrt[5]{7}}{140} + \frac{\sqrt{5} \sqrt[5]{7}}{140} + \frac{\left(-1\right) \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{35}\right) \log{\left(\operatorname{y'}{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)} - 2 \left(- \frac{\sqrt[5]{7}}{140} + \frac{\sqrt{5} \sqrt[5]{7}}{140} + \frac{\sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}}}{35}\right) \log{\left(\operatorname{y'}{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)} - 2 \left(- \frac{\sqrt{5} \sqrt[5]{7}}{140} - \frac{\sqrt[5]{7}}{140} + \frac{\left(-1\right) \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{35}\right) \log{\left(\operatorname{y'}{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)} - 2 \left(- \frac{\sqrt{5} \sqrt[5]{7}}{140} - \frac{\sqrt[5]{7}}{140} + \frac{\sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{35}\right) \log{\left(\operatorname{y'}{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)} = C_{1} - \int \frac{1}{y{\left(x \right)}}\, dx$$
tomemos estas integrales
$$\operatorname{y_{1}} = \int \left(- 2 \operatorname{RootSum} {\left(7503125 t^{5} - 1, \left( t \mapsto t \log{\left(35 t + \frac{d}{d x} y{\left(x \right)} \right)} \right)\right)}\right)\, dx = \int \left(C_{1} - \int \frac{1}{y{\left(x \right)}}\, dx\right)\, dx$$ =
$$\operatorname{y_{1}} = - \frac{\sqrt[5]{7} \left(\int \sqrt{5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\, dx + \int \sqrt{5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\, dx + \int \left(- \sqrt{5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx + \int \left(- \sqrt{5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx + \int \left(- \sqrt{2} i \sqrt{5 - \sqrt{5}} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx + \int \sqrt{2} i \sqrt{5 - \sqrt{5}} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\, dx + \int \left(- \sqrt{2} i \sqrt{\sqrt{5} + 5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\right)\, dx + \int \sqrt{2} i \sqrt{\sqrt{5} + 5} \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\, dx + \int 4 \log{\left(\frac{d}{d x} y{\left(x \right)} + \sqrt[5]{7} \right)}\, dx + \int \left(- \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\right)\, dx + \int \left(- \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt[5]{7}}{4} + \frac{\sqrt{5} \sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{\sqrt{5}}{8} + \frac{5}{8}} \right)}\right)\, dx + \int \left(- \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} - \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx + \int \left(- \log{\left(\frac{d}{d x} y{\left(x \right)} - \frac{\sqrt{5} \sqrt[5]{7}}{4} - \frac{\sqrt[5]{7}}{4} + \sqrt[5]{7} i \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}} \right)}\right)\, dx\right)}{70} = C_{2} + \int \left(C_{1} - \int \frac{1}{y{\left(x \right)}}\, dx\right)\, dx$$
Clasificación
factorable