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Ecuaciones diferenciales/ dy/dx-y=-cos(x)y^2

Ecuación diferencial dy/dx-y=-cos(x)y^2

El profesor se sorprenderá mucho al ver tu solución correcta😉

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Para el problema de Cauchy:

y() =
y'() =
y''() =
y'''() =
y''''() =

Gráfico:

interior superior
⌨

Solución

Ha introducido [src] 
        d            2          
-y(x) + --(y(x)) = -y (x)*cos(x)
        dx
$$- y{\left(x \right)} + \frac{d}{d x} y{\left(x \right)} = - y^{2}{\left(x \right)} \cos{\left(x \right)}$$ 
-y + y' = -y^2*cos(x)
Respuesta [src] 
                                3 /              /          2              2\\       4 /                         /                  /          2              2\                            \                \       5 /         /          2              2\            /      2                     /                  /          2              2\                            \                                   /              2       2                                  \\                                                      \       2                            
                            C1*x *\C1 + (1 - C1)*\(1 - 2*C1)  - 2*C1 + 2*C1 //   C1*x *\C1*(1 - 2*C1) + (1 - C1)*\2*C1 + (1 - 2*C1)*\(1 - 2*C1)  - 2*C1 + 2*C1 / + 6*C1*(1 - C1)*(-1 + 2*C1)/ + 4*C1*(1 - C1)/   C1*x *\-C1 + C1*\(1 - 2*C1)  - 2*C1 + 2*C1 / + (1 - C1)*\- 6*C1  + 4*C1 + (1 - 2*C1)*\2*C1 + (1 - 2*C1)*\(1 - 2*C1)  - 2*C1 + 2*C1 / + 6*C1*(1 - C1)*(-1 + 2*C1)/ + 6*C1*(1 - 2*C1) + 2*C1*(1 - C1)*\1 - (1 - 2*C1)  - 8*C1  + 8*C1 + 6*(1 - 2*C1)*(-1 + 2*C1)// + 2*C1*(1 - C1)*(5 - 9*C1) + 6*C1*(1 - C1)*(1 - 2*C1)/   C1*x *(1 - C1)*(1 - 2*C1)    / 6\
y(x) = C1 + C1*x*(1 - C1) + -------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------- + O\x /
                                                    6                                                                                          24                                                                                                                                                                                                                          120                                                                                                                                                                         2
$$y{\left(x \right)} = C_{1} + C_{1} x \left(1 - C_{1}\right) + \frac{C_{1} x^{2} \left(1 - 2 C_{1}\right) \left(1 - C_{1}\right)}{2} + \frac{C_{1} x^{3} \left(C_{1} + \left(1 - C_{1}\right) \left(2 C_{1}^{2} - 2 C_{1} + \left(1 - 2 C_{1}\right)^{2}\right)\right)}{6} + \frac{C_{1} x^{4} \left(C_{1} \left(1 - 2 C_{1}\right) + 4 C_{1} \left(1 - C_{1}\right) + \left(1 - C_{1}\right) \left(6 C_{1} \left(1 - C_{1}\right) \left(2 C_{1} - 1\right) + 2 C_{1} + \left(1 - 2 C_{1}\right) \left(2 C_{1}^{2} - 2 C_{1} + \left(1 - 2 C_{1}\right)^{2}\right)\right)\right)}{24} + \frac{C_{1} x^{5} \left(6 C_{1} \left(1 - 2 C_{1}\right) \left(1 - C_{1}\right) + 2 C_{1} \left(1 - C_{1}\right) \left(5 - 9 C_{1}\right) + C_{1} \left(2 C_{1}^{2} - 2 C_{1} + \left(1 - 2 C_{1}\right)^{2}\right) - C_{1} + \left(1 - C_{1}\right) \left(- 6 C_{1}^{2} + 6 C_{1} \left(1 - 2 C_{1}\right) + 2 C_{1} \left(1 - C_{1}\right) \left(- 8 C_{1}^{2} + 8 C_{1} - \left(1 - 2 C_{1}\right)^{2} + 6 \left(1 - 2 C_{1}\right) \left(2 C_{1} - 1\right) + 1\right) + 4 C_{1} + \left(1 - 2 C_{1}\right) \left(6 C_{1} \left(1 - C_{1}\right) \left(2 C_{1} - 1\right) + 2 C_{1} + \left(1 - 2 C_{1}\right) \left(2 C_{1}^{2} - 2 C_{1} + \left(1 - 2 C_{1}\right)^{2}\right)\right)\right)\right)}{120} + O\left(x^{6}\right)$$
Trazar el gráfico con C=-1
Trazar el gráfico con C=0
Trazar el gráfico con C=1
Gráfico para el problema de Cauchy
Clasificación
1st power series
lie group
Respuesta numérica [src] 
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 955417393.6655939)
(-5.555555555555555, 2.17e-322)
(-3.333333333333333, nan)
(-1.1111111111111107, 2.78363573e-315)
(1.1111111111111107, 8.427456047434801e+197)
(3.333333333333334, 3.1933833808213433e-248)
(5.555555555555557, 2.957741787671986e-32)
(7.777777777777779, 8.388243566974613e+296)
(10.0, 3.861029683e-315)
(10.0, 3.861029683e-315)
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