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Ecuación diferencial y’’*(y-4)^2=y’*(y’-4)^4
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Solución
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2 4
2 d / d \ d
(-4 + y(x)) *---(y(x)) = |-4 + --(y(x))| *--(y(x))
2 \ dx / dx
dx
$$\left(y{\left(x \right)} - 4\right)^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} = \left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}$$
(y - 4)^2*y'' = (y' - 4)^4*y'
Solución detallada
Tenemos la ecuación:
$$\left(y{\left(x \right)} - 4\right)^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} = \left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}$$
Esta ecuación diferencial tiene la forma:
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y' \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = - \frac{1}{\left(y{\left(x \right)} - 4\right)^{2}}$$
$$\operatorname{g_{2}}{\left(y' \right)} = - \left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}$$
Pasemos la ecuación a la forma:
Dividamos ambos miembros de la ecuación en g2(y')
$$- \left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}$$
obtendremos
$$- \frac{\frac{d^{2}}{d x^{2}} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}} = - \frac{1}{\left(y{\left(x \right)} - 4\right)^{2}}$$
Con esto hemos separado las variables x y y'.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$- \frac{dx \frac{d^{2}}{d x^{2}} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}} = - \frac{dx}{\left(y{\left(x \right)} - 4\right)^{2}}$$
o
$$- \frac{dy'}{\left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}} = - \frac{dx}{\left(y{\left(x \right)} - 4\right)^{2}}$$
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.
$$\int \left(- \frac{1}{y' \left(y' - 4\right)^{4}}\right)\, dy' = \int \left(- \frac{1}{\left(y{\left(x \right)} - 4\right)^{2}}\right)\, dx$$
Tomemos estas integrales
$$\frac{3 y'^{2} - 30 y' + 88}{192 y'^{3} - 2304 y'^{2} + 9216 y' - 12288} - \frac{\log{\left(y' \right)}}{256} + \frac{\log{\left(y' - 4 \right)}}{256} = Const - \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx$$
Hemos recibido una ecuación ordinaria con la incógnica y'.
(Const - es una constante)
La solución:
$$\operatorname{y'1} = \frac{3 \operatorname{y'}^{2}{\left(x \right)} - 30 \operatorname{y'}{\left(x \right)} + 88}{192 \left(\operatorname{y'}^{3}{\left(x \right)} - 12 \operatorname{y'}^{2}{\left(x \right)} + 48 \operatorname{y'}{\left(x \right)} - 64\right)} + \frac{\log{\left(\operatorname{y'}{\left(x \right)} - 4 \right)}}{256} - \frac{\log{\left(\operatorname{y'}{\left(x \right)} \right)}}{256} + \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx = C_{1}$$
tomemos estas integrales
$$\operatorname{y_{1}} = \int \left(\frac{3 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} - 30 \frac{d}{d x} y{\left(x \right)} + 88}{192 \left(\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64\right)} + \frac{\log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)}}{256} - \frac{\log{\left(\frac{d}{d x} y{\left(x \right)} \right)}}{256} + \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx\right)\, dx = \int C_{1}\, dx$$ =
$$\operatorname{y_{1}} = \frac{\int \left(- \frac{192 \log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{192 \log{\left(\frac{d}{d x} y{\left(x \right)} \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{120 \frac{d}{d x} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{49152 \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{144 \log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)} \frac{d}{d x} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{36 \log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)} \left(\frac{d}{d x} y{\left(x \right)}\right)^{2}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{3 \log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)} \left(\frac{d}{d x} y{\left(x \right)}\right)^{3}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{144 \log{\left(\frac{d}{d x} y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{36 \log{\left(\frac{d}{d x} y{\left(x \right)} \right)} \left(\frac{d}{d x} y{\left(x \right)}\right)^{2}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{3 \log{\left(\frac{d}{d x} y{\left(x \right)} \right)} \left(\frac{d}{d x} y{\left(x \right)}\right)^{3}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{36864 \frac{d}{d x} y{\left(x \right)} \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{9216 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{768 \left(\frac{d}{d x} y{\left(x \right)}\right)^{3} \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \frac{352}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx}{768} = C_{1} x + C_{2}$$
$$\left(y{\left(x \right)} - 4\right)^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} = \left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}$$
Esta ecuación diferencial tiene la forma:
f1(x)*g1(y')*y'' = f2(x)*g2(y'),
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y' \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = - \frac{1}{\left(y{\left(x \right)} - 4\right)^{2}}$$
$$\operatorname{g_{2}}{\left(y' \right)} = - \left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}$$
Pasemos la ecuación a la forma:
g1(y')/g2(y')*y''= f2(x)/f1(x).
Dividamos ambos miembros de la ecuación en g2(y')
$$- \left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}$$
obtendremos
$$- \frac{\frac{d^{2}}{d x^{2}} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}} = - \frac{1}{\left(y{\left(x \right)} - 4\right)^{2}}$$
Con esto hemos separado las variables x y y'.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$- \frac{dx \frac{d^{2}}{d x^{2}} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}} = - \frac{dx}{\left(y{\left(x \right)} - 4\right)^{2}}$$
o
$$- \frac{dy'}{\left(\frac{d}{d x} y{\left(x \right)} - 4\right)^{4} \frac{d}{d x} y{\left(x \right)}} = - \frac{dx}{\left(y{\left(x \right)} - 4\right)^{2}}$$
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.
$$\int \left(- \frac{1}{y' \left(y' - 4\right)^{4}}\right)\, dy' = \int \left(- \frac{1}{\left(y{\left(x \right)} - 4\right)^{2}}\right)\, dx$$
Tomemos estas integrales
$$\frac{3 y'^{2} - 30 y' + 88}{192 y'^{3} - 2304 y'^{2} + 9216 y' - 12288} - \frac{\log{\left(y' \right)}}{256} + \frac{\log{\left(y' - 4 \right)}}{256} = Const - \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx$$
Hemos recibido una ecuación ordinaria con la incógnica y'.
(Const - es una constante)
La solución:
$$\operatorname{y'1} = \frac{3 \operatorname{y'}^{2}{\left(x \right)} - 30 \operatorname{y'}{\left(x \right)} + 88}{192 \left(\operatorname{y'}^{3}{\left(x \right)} - 12 \operatorname{y'}^{2}{\left(x \right)} + 48 \operatorname{y'}{\left(x \right)} - 64\right)} + \frac{\log{\left(\operatorname{y'}{\left(x \right)} - 4 \right)}}{256} - \frac{\log{\left(\operatorname{y'}{\left(x \right)} \right)}}{256} + \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx = C_{1}$$
tomemos estas integrales
$$\operatorname{y_{1}} = \int \left(\frac{3 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} - 30 \frac{d}{d x} y{\left(x \right)} + 88}{192 \left(\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64\right)} + \frac{\log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)}}{256} - \frac{\log{\left(\frac{d}{d x} y{\left(x \right)} \right)}}{256} + \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx\right)\, dx = \int C_{1}\, dx$$ =
$$\operatorname{y_{1}} = \frac{\int \left(- \frac{192 \log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{192 \log{\left(\frac{d}{d x} y{\left(x \right)} \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{120 \frac{d}{d x} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{49152 \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{144 \log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)} \frac{d}{d x} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{36 \log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)} \left(\frac{d}{d x} y{\left(x \right)}\right)^{2}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{3 \log{\left(\frac{d}{d x} y{\left(x \right)} - 4 \right)} \left(\frac{d}{d x} y{\left(x \right)}\right)^{3}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{144 \log{\left(\frac{d}{d x} y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{36 \log{\left(\frac{d}{d x} y{\left(x \right)} \right)} \left(\frac{d}{d x} y{\left(x \right)}\right)^{2}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{3 \log{\left(\frac{d}{d x} y{\left(x \right)} \right)} \left(\frac{d}{d x} y{\left(x \right)}\right)^{3}}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{36864 \frac{d}{d x} y{\left(x \right)} \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \left(- \frac{9216 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\right)\, dx + \int \frac{768 \left(\frac{d}{d x} y{\left(x \right)}\right)^{3} \int \frac{1}{y^{2}{\left(x \right)} - 8 y{\left(x \right)} + 16}\, dx}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx + \int \frac{352}{\left(\frac{d}{d x} y{\left(x \right)}\right)^{3} - 12 \left(\frac{d}{d x} y{\left(x \right)}\right)^{2} + 48 \frac{d}{d x} y{\left(x \right)} - 64}\, dx}{768} = C_{1} x + C_{2}$$