Sr Examen
Ecuación diferencial (1-2xy)y'=y(y-1)
El profesor se sorprenderá mucho al ver tu solución correcta😉
⌨
Solución
Ha introducido
[src]
d
(1 - 2*x*y(x))*--(y(x)) = (-1 + y(x))*y(x)
dx
$$\left(- 2 x y{\left(x \right)} + 1\right) \frac{d}{d x} y{\left(x \right)} = \left(y{\left(x \right)} - 1\right) y{\left(x \right)}$$
(-2*x*y + 1)*y' = (y - 1)*y
Respuesta
[src]
2 / 2 \ 3 / 2 3 / 2 2\ \ 4 / 3 4 / 2 3 / 2 2\ \ / 2 2\ / 2 \ 2 2 2 \ 5 / 4 5 / 3 4 / 2 3 / 2 2\ \ 2 2 / 2 2 \ / 2 2\ / 2 \ 2 / 2 \ 2 2 2 \ / 2 3 / 2 2\ \ / 2 3 / 2 2\ / 2 \ 2 / 2 \ \ 2 / 2 2\ 2 / 2 2 \ 2 / 2 \ 3 3 3 3 \
C1*x *\-2*C1 + 2*C1 + (-1 + C1)*(-1 + 2*C1)/ C1*x *\- 8*C1 + 8*C1 + (-1 + C1)*\(-1 + 2*C1) - 6*C1 + 8*C1 / + 2*C1*(-1 + C1)*(-1 + 2*C1) + 2*C1*(-1 + C1)*(-2 + 3*C1)/ C1*x *\- 48*C1 + 48*C1 + (-1 + C1)*\- 28*C1 + 36*C1 + (-1 + 2*C1)*\(-1 + 2*C1) - 6*C1 + 8*C1 / + 2*C1*(-1 + C1)*(-5 + 12*C1) + 4*C1*(-1 + C1)*(-1 + 3*C1) + 4*C1*(-1 + 2*C1)*(-2 + 3*C1)/ + 2*C1*(-1 + C1)*\(-1 + 2*C1) - 6*C1 + 8*C1 / + 4*C1*(-1 + C1)*\-7*C1 + 9*C1 + (-1 + C1)*(-1 + 3*C1) + (-1 + 2*C1)*(-2 + 3*C1)/ + 8*C1 *(-1 + C1)*(-1 + 2*C1) + 8*C1 *(-1 + C1)*(-3 + 4*C1) + 8*C1 *(-1 + C1)*(-2 + 3*C1)/ C1*x *\- 384*C1 + 384*C1 + (-1 + C1)*\- 208*C1 + 256*C1 + (-1 + 2*C1)*\- 28*C1 + 36*C1 + (-1 + 2*C1)*\(-1 + 2*C1) - 6*C1 + 8*C1 / + 2*C1*(-1 + C1)*(-5 + 12*C1) + 4*C1*(-1 + C1)*(-1 + 3*C1) + 4*C1*(-1 + 2*C1)*(-2 + 3*C1)/ + 8*C1 *(-2 + 3*C1) + 2*C1*(-1 + C1)*\(-1 + 2*C1) - 60*C1 + 92*C1 + 2*(-1 + 2*C1)*(-5 + 12*C1) + 6*(-1 + 2*C1)*(-1 + 3*C1)/ + 2*C1*(-2 + 3*C1)*\(-1 + 2*C1) - 6*C1 + 8*C1 / + 4*C1*(-1 + 2*C1)*\-7*C1 + 9*C1 + (-1 + C1)*(-1 + 3*C1) + (-1 + 2*C1)*(-2 + 3*C1)/ + 4*C1 *(-1 + C1)*(-5 + 12*C1) + 12*C1*(-1 + C1)*\-7*C1 + 12*C1 + (-1 + 2*C1)*(-1 + 3*C1)/ + 16*C1 *(-1 + C1)*(-1 + 3*C1) + 16*C1 *(-1 + 2*C1)*(-3 + 4*C1) + 48*C1 *(-1 + C1)*(-1 + 2*C1)/ + 2*C1*(-1 + C1)*\- 28*C1 + 36*C1 + (-1 + 2*C1)*\(-1 + 2*C1) - 6*C1 + 8*C1 / + 2*C1*(-1 + C1)*(-5 + 12*C1) + 4*C1*(-1 + C1)*(-1 + 3*C1) + 4*C1*(-1 + 2*C1)*(-2 + 3*C1)/ + 2*C1*(-1 + C1)*\- 104*C1 + 128*C1 + (-2 + 3*C1)*\(-1 + 2*C1) - 6*C1 + 8*C1 / + 2*(-1 + 2*C1)*\-7*C1 + 9*C1 + (-1 + C1)*(-1 + 3*C1) + (-1 + 2*C1)*(-2 + 3*C1)/ + 4*C1*(-2 + 3*C1) + 6*(-1 + C1)*\-7*C1 + 12*C1 + (-1 + 2*C1)*(-1 + 3*C1)/ + 2*C1*(-1 + C1)*(-5 + 12*C1) + 8*C1*(-1 + C1)*(-1 + 3*C1) + 8*C1*(-1 + 2*C1)*(-3 + 4*C1) + 24*C1*(-1 + C1)*(-1 + 2*C1)/ + 8*C1 *(-1 + C1)*\(-1 + 2*C1) - 6*C1 + 8*C1 / + 8*C1 *(-1 + C1)*\(-2 + 3*C1) - 26*C1 + 32*C1 + 2*(-1 + C1)*(-1 + 3*C1) + 2*(-1 + 2*C1)*(-3 + 4*C1) + 6*(-1 + C1)*(-1 + 2*C1)/ + 16*C1 *(-1 + C1)*\-7*C1 + 9*C1 + (-1 + C1)*(-1 + 3*C1) + (-1 + 2*C1)*(-2 + 3*C1)/ + 48*C1 *(-1 + C1)*(-1 + 2*C1) + 48*C1 *(-1 + C1)*(-4 + 5*C1) + 48*C1 *(-1 + C1)*(-3 + 4*C1) + 48*C1 *(-1 + C1)*(-2 + 3*C1)/ / 6\
y(x) = C1 + C1*x*(-1 + C1) + --------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + O\x /
2 6 24 120
$$y{\left(x \right)} = C_{1} + C_{1} x \left(C_{1} - 1\right) + \frac{C_{1} x^{2} \left(2 C_{1}^{2} - 2 C_{1} + \left(C_{1} - 1\right) \left(2 C_{1} - 1\right)\right)}{2} + \frac{C_{1} x^{3} \left(8 C_{1}^{3} - 8 C_{1}^{2} + 2 C_{1} \left(C_{1} - 1\right) \left(2 C_{1} - 1\right) + 2 C_{1} \left(C_{1} - 1\right) \left(3 C_{1} - 2\right) + \left(C_{1} - 1\right) \left(8 C_{1}^{2} - 6 C_{1} + \left(2 C_{1} - 1\right)^{2}\right)\right)}{6} + \frac{C_{1} x^{4} \left(48 C_{1}^{4} - 48 C_{1}^{3} + 8 C_{1}^{2} \left(C_{1} - 1\right) \left(2 C_{1} - 1\right) + 8 C_{1}^{2} \left(C_{1} - 1\right) \left(3 C_{1} - 2\right) + 8 C_{1}^{2} \left(C_{1} - 1\right) \left(4 C_{1} - 3\right) + 2 C_{1} \left(C_{1} - 1\right) \left(8 C_{1}^{2} - 6 C_{1} + \left(2 C_{1} - 1\right)^{2}\right) + 4 C_{1} \left(C_{1} - 1\right) \left(9 C_{1}^{2} - 7 C_{1} + \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + \left(2 C_{1} - 1\right) \left(3 C_{1} - 2\right)\right) + \left(C_{1} - 1\right) \left(36 C_{1}^{3} - 28 C_{1}^{2} + 4 C_{1} \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + 2 C_{1} \left(C_{1} - 1\right) \left(12 C_{1} - 5\right) + 4 C_{1} \left(2 C_{1} - 1\right) \left(3 C_{1} - 2\right) + \left(2 C_{1} - 1\right) \left(8 C_{1}^{2} - 6 C_{1} + \left(2 C_{1} - 1\right)^{2}\right)\right)\right)}{24} + \frac{C_{1} x^{5} \left(384 C_{1}^{5} - 384 C_{1}^{4} + 48 C_{1}^{3} \left(C_{1} - 1\right) \left(2 C_{1} - 1\right) + 48 C_{1}^{3} \left(C_{1} - 1\right) \left(3 C_{1} - 2\right) + 48 C_{1}^{3} \left(C_{1} - 1\right) \left(4 C_{1} - 3\right) + 48 C_{1}^{3} \left(C_{1} - 1\right) \left(5 C_{1} - 4\right) + 8 C_{1}^{2} \left(C_{1} - 1\right) \left(8 C_{1}^{2} - 6 C_{1} + \left(2 C_{1} - 1\right)^{2}\right) + 16 C_{1}^{2} \left(C_{1} - 1\right) \left(9 C_{1}^{2} - 7 C_{1} + \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + \left(2 C_{1} - 1\right) \left(3 C_{1} - 2\right)\right) + 8 C_{1}^{2} \left(C_{1} - 1\right) \left(32 C_{1}^{2} - 26 C_{1} + 6 \left(C_{1} - 1\right) \left(2 C_{1} - 1\right) + 2 \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + 2 \left(2 C_{1} - 1\right) \left(4 C_{1} - 3\right) + \left(3 C_{1} - 2\right)^{2}\right) + 2 C_{1} \left(C_{1} - 1\right) \left(36 C_{1}^{3} - 28 C_{1}^{2} + 4 C_{1} \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + 2 C_{1} \left(C_{1} - 1\right) \left(12 C_{1} - 5\right) + 4 C_{1} \left(2 C_{1} - 1\right) \left(3 C_{1} - 2\right) + \left(2 C_{1} - 1\right) \left(8 C_{1}^{2} - 6 C_{1} + \left(2 C_{1} - 1\right)^{2}\right)\right) + 2 C_{1} \left(C_{1} - 1\right) \left(128 C_{1}^{3} - 104 C_{1}^{2} + 24 C_{1} \left(C_{1} - 1\right) \left(2 C_{1} - 1\right) + 8 C_{1} \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + 2 C_{1} \left(C_{1} - 1\right) \left(12 C_{1} - 5\right) + 8 C_{1} \left(2 C_{1} - 1\right) \left(4 C_{1} - 3\right) + 4 C_{1} \left(3 C_{1} - 2\right)^{2} + 6 \left(C_{1} - 1\right) \left(12 C_{1}^{2} - 7 C_{1} + \left(2 C_{1} - 1\right) \left(3 C_{1} - 1\right)\right) + 2 \left(2 C_{1} - 1\right) \left(9 C_{1}^{2} - 7 C_{1} + \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + \left(2 C_{1} - 1\right) \left(3 C_{1} - 2\right)\right) + \left(3 C_{1} - 2\right) \left(8 C_{1}^{2} - 6 C_{1} + \left(2 C_{1} - 1\right)^{2}\right)\right) + \left(C_{1} - 1\right) \left(256 C_{1}^{4} - 208 C_{1}^{3} + 48 C_{1}^{2} \left(C_{1} - 1\right) \left(2 C_{1} - 1\right) + 16 C_{1}^{2} \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + 4 C_{1}^{2} \left(C_{1} - 1\right) \left(12 C_{1} - 5\right) + 16 C_{1}^{2} \left(2 C_{1} - 1\right) \left(4 C_{1} - 3\right) + 8 C_{1}^{2} \left(3 C_{1} - 2\right)^{2} + 12 C_{1} \left(C_{1} - 1\right) \left(12 C_{1}^{2} - 7 C_{1} + \left(2 C_{1} - 1\right) \left(3 C_{1} - 1\right)\right) + 2 C_{1} \left(C_{1} - 1\right) \left(92 C_{1}^{2} - 60 C_{1} + \left(2 C_{1} - 1\right)^{2} + 6 \left(2 C_{1} - 1\right) \left(3 C_{1} - 1\right) + 2 \left(2 C_{1} - 1\right) \left(12 C_{1} - 5\right)\right) + 4 C_{1} \left(2 C_{1} - 1\right) \left(9 C_{1}^{2} - 7 C_{1} + \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + \left(2 C_{1} - 1\right) \left(3 C_{1} - 2\right)\right) + 2 C_{1} \left(3 C_{1} - 2\right) \left(8 C_{1}^{2} - 6 C_{1} + \left(2 C_{1} - 1\right)^{2}\right) + \left(2 C_{1} - 1\right) \left(36 C_{1}^{3} - 28 C_{1}^{2} + 4 C_{1} \left(C_{1} - 1\right) \left(3 C_{1} - 1\right) + 2 C_{1} \left(C_{1} - 1\right) \left(12 C_{1} - 5\right) + 4 C_{1} \left(2 C_{1} - 1\right) \left(3 C_{1} - 2\right) + \left(2 C_{1} - 1\right) \left(8 C_{1}^{2} - 6 C_{1} + \left(2 C_{1} - 1\right)^{2}\right)\right)\right)\right)}{120} + O\left(x^{6}\right)$$
Gráfico para el problema de Cauchy
Clasificación
factorable
1st exact
1st power series
lie group
1st exact Integral
Respuesta numérica
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(x, y):
(0.0, 0.75)
(0.4444444444444444, 0.6066664971477642)
(0.8888888888888888, 0.3323580175978124)
(1.3333333333333333, 0.16482181461747109)
(1.7777777777777777, 0.088292922702926)
(2.2222222222222223, 0.05016919663125771)
(2.6666666666666665, 0.02962763646915432)
(3.1111111111111107, 0.017950178492242147)
(3.5555555555555554, 0.011065548849617513)
(4.0, 0.006903397036803458)
(4.0, 0.006903397036803458)