Sr Examen
Ecuación diferencial (1+сosx)/(x+sinx)*dx-((3y-5)^3)dy=0
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Solución
Ha introducido
[src]
1 d cos(x) d 3 d 2 d ---------- + 125*--(y(x)) + ---------- - 225*--(y(x))*y(x) - 27*y (x)*--(y(x)) + 135*y (x)*--(y(x)) = 0 x + sin(x) dx x + sin(x) dx dx dx
$$- 27 y^{3}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 135 y^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} - 225 y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 125 \frac{d}{d x} y{\left(x \right)} + \frac{\cos{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{1}{x + \sin{\left(x \right)}} = 0$$
-27*y^3*y' + 135*y^2*y' - 225*y*y' + 125*y' + cos(x)/(x + sin(x)) + 1/(x + sin(x)) = 0
Solución detallada
Tenemos la ecuación:
$$- 27 y^{3}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 135 y^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} - 225 y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 125 \frac{d}{d x} y{\left(x \right)} + \frac{\cos{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{1}{x + \sin{\left(x \right)}} = 0$$
Esta ecuación diferencial tiene la forma:
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = \frac{\cos{\left(x \right)} + 1}{x + \sin{\left(x \right)}}$$
$$\operatorname{g_{2}}{\left(y \right)} = \frac{1}{\left(3 y{\left(x \right)} - 5\right)^{3}}$$
Pasemos la ecuación a la forma:
Dividamos ambos miembros de la ecuación en g2(y)
$$\frac{1}{\left(3 y{\left(x \right)} - 5\right)^{3}}$$
obtendremos
$$\left(3 y{\left(x \right)} - 5\right)^{3} \frac{d}{d x} y{\left(x \right)} = \frac{\cos{\left(x \right)} + 1}{x + \sin{\left(x \right)}}$$
Con esto hemos separado las variables x y y.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$dx \left(3 y{\left(x \right)} - 5\right)^{3} \frac{d}{d x} y{\left(x \right)} = \frac{dx \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}}$$
o
$$dy \left(3 y{\left(x \right)} - 5\right)^{3} = \frac{dx \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}}$$
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.
$$\int \left(3 y - 5\right)^{3}\, dy = \int \frac{\cos{\left(x \right)} + 1}{x + \sin{\left(x \right)}}\, dx$$
Tomemos estas integrales
$$\frac{27 y^{4}}{4} - 45 y^{3} + \frac{225 y^{2}}{2} - 125 y = Const + \log{\left(x + \sin{\left(x \right)} \right)}$$
Hemos recibido una ecuación ordinaria con la incógnica y.
(Const - es una constante)
La solución:
$$\operatorname{y_{1}} = y{\left(x \right)} = \frac{5}{3} - \frac{\sqrt{- \sqrt{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}}{3}$$
$$\operatorname{y_{2}} = y{\left(x \right)} = \frac{5}{3} - \frac{\sqrt[4]{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}{3}$$
$$\operatorname{y_{3}} = y{\left(x \right)} = \frac{\sqrt{- \sqrt{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}}{3} + \frac{5}{3}$$
$$\operatorname{y_{4}} = y{\left(x \right)} = \frac{\sqrt[4]{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}{3} + \frac{5}{3}$$
$$- 27 y^{3}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 135 y^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} - 225 y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 125 \frac{d}{d x} y{\left(x \right)} + \frac{\cos{\left(x \right)}}{x + \sin{\left(x \right)}} + \frac{1}{x + \sin{\left(x \right)}} = 0$$
Esta ecuación diferencial tiene la forma:
f1(x)*g1(y)*y' = f2(x)*g2(y),
donde
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = \frac{\cos{\left(x \right)} + 1}{x + \sin{\left(x \right)}}$$
$$\operatorname{g_{2}}{\left(y \right)} = \frac{1}{\left(3 y{\left(x \right)} - 5\right)^{3}}$$
Pasemos la ecuación a la forma:
g1(y)/g2(y)*y'= f2(x)/f1(x).
Dividamos ambos miembros de la ecuación en g2(y)
$$\frac{1}{\left(3 y{\left(x \right)} - 5\right)^{3}}$$
obtendremos
$$\left(3 y{\left(x \right)} - 5\right)^{3} \frac{d}{d x} y{\left(x \right)} = \frac{\cos{\left(x \right)} + 1}{x + \sin{\left(x \right)}}$$
Con esto hemos separado las variables x y y.
Ahora multipliquemos las dos partes de la ecuación por dx,
entonces la ecuación será así
$$dx \left(3 y{\left(x \right)} - 5\right)^{3} \frac{d}{d x} y{\left(x \right)} = \frac{dx \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}}$$
o
$$dy \left(3 y{\left(x \right)} - 5\right)^{3} = \frac{dx \left(\cos{\left(x \right)} + 1\right)}{x + \sin{\left(x \right)}}$$
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.
$$\int \left(3 y - 5\right)^{3}\, dy = \int \frac{\cos{\left(x \right)} + 1}{x + \sin{\left(x \right)}}\, dx$$
Tomemos estas integrales
$$\frac{27 y^{4}}{4} - 45 y^{3} + \frac{225 y^{2}}{2} - 125 y = Const + \log{\left(x + \sin{\left(x \right)} \right)}$$
Hemos recibido una ecuación ordinaria con la incógnica y.
(Const - es una constante)
La solución:
$$\operatorname{y_{1}} = y{\left(x \right)} = \frac{5}{3} - \frac{\sqrt{- \sqrt{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}}{3}$$
$$\operatorname{y_{2}} = y{\left(x \right)} = \frac{5}{3} - \frac{\sqrt[4]{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}{3}$$
$$\operatorname{y_{3}} = y{\left(x \right)} = \frac{\sqrt{- \sqrt{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}}{3} + \frac{5}{3}$$
$$\operatorname{y_{4}} = y{\left(x \right)} = \frac{\sqrt[4]{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}{3} + \frac{5}{3}$$
Respuesta
[src]
______________________________
/ _________________________
5 \/ -\/ C1 + 12*log(x + sin(x))
y(x) = - - ---------------------------------
3 3
$$y{\left(x \right)} = \frac{5}{3} - \frac{\sqrt{- \sqrt{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}}{3}$$
4 _________________________
5 \/ C1 + 12*log(x + sin(x))
y(x) = - - ---------------------------
3 3
$$y{\left(x \right)} = \frac{5}{3} - \frac{\sqrt[4]{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}{3}$$
______________________________
/ _________________________
5 \/ -\/ C1 + 12*log(x + sin(x))
y(x) = - + ---------------------------------
3 3
$$y{\left(x \right)} = \frac{\sqrt{- \sqrt{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}}{3} + \frac{5}{3}$$
4 _________________________
5 \/ C1 + 12*log(x + sin(x))
y(x) = - + ---------------------------
3 3
$$y{\left(x \right)} = \frac{\sqrt[4]{C_{1} + 12 \log{\left(x + \sin{\left(x \right)} \right)}}}{3} + \frac{5}{3}$$
Gráfico para el problema de Cauchy
Clasificación
factorable
separable
1st exact
lie group
separable Integral
1st exact Integral
Respuesta numérica
[src]
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.7536158373619442)
(-5.555555555555555, 0.7834965868087012)
(-3.333333333333333, 0.8082958121892424)
(-1.1111111111111107, 0.8358480098872121)
(1.1111111111111107, 1.6666663891186553)
(3.333333333333334, 3.1933833808213433e-248)
(5.555555555555557, 4.32563549789618e-37)
(7.777777777777779, 8.388243571828907e+296)
(10.0, 3.861029683e-315)
(10.0, 3.861029683e-315)