Sr Examen
Ecuación diferencial 1-2xyy'=y^3(y')
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
[src]
d 3 d
1 - 2*x*--(y(x))*y(x) = y (x)*--(y(x))
dx dx
$$- 2 x y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 1 = y^{3}{\left(x \right)} \frac{d}{d x} y{\left(x \right)}$$
-2*x*y*y' + 1 = y^3*y'
Respuesta
[src]
/ / / 21\ / 14\\ / / 14\ / 21\ / 18\\ \
| | 3*|10 + ---| 12*|-5 - ---|| | 18*|-5 - ---| 3*|10 + ---| 21*|5 + ---|| |
| | | 2| | 2|| | | 2| | 2| | 2|| |
/ / 21\ / 14\ \ | / 18\ / 14\ | 144 \ C1 / \ C1 /| | 360 \ C1 / \ C1 / \ C1 /| / 21\ / 45\ / / 21\ / 14\\ / / 18\ / 14\ / 21\ / 45\\ |
| 3*|10 + ---| 12*|-5 - ---| | | 24*|-7 - ---| 24*|-5 - ---| 3*|-56 - --- - ------------ + -------------| 4*|112 + --- - ------------- + ------------ + ------------| 10*|10 + ---| 32*|14 + ---| | 3*|10 + ---| 12*|-5 - ---|| | 12*|-7 - ---| 12*|-5 - ---| 5*|10 + ---| 16*|14 + ---|| |
| | 2| | 2| | | | 2| | 2| | 2 2 2 | | 2 2 2 2 | | 2| | 2| | | 2| | 2|| | | 2| | 2| | 2| | 2|| |
/ 21\ | 144 \ C1 / \ C1 / / 21\ / 18\| | 1320 \ C1 / \ C1 / \ C1 C1 C1 / \ C1 C1 C1 C1 / \ C1 / \ C1 / / 18\ | 144 \ C1 / \ C1 /| | 660 \ C1 / \ C1 / \ C1 / \ C1 /| / 21\ / 55\|
| 10 + ---| | -56 - --- - ------------ + ------------- 2*|10 + ---| 8*|-7 - ---|| | 432 + ---- - ------------- - ------------- - -------------------------------------------- + ----------------------------------------------------------- + ------------- + ------------- 32*|-7 - ---| 2*|-56 - --- - ------------ + -------------| 2*|216 + --- - ------------- - ------------- + ------------ + -------------| 8*|10 + ---| 24*|18 + ---||
| 2| | 2 2 2 | 2| | 2|| | 2 2 2 2 2 2 2 | 2| | 2 2 2 | | 2 2 2 2 2 | | 2| | 2||
2 / 3 \ 3 | 16 C1 | 4 | 120 C1 C1 C1 \ C1 / \ C1 /| 5 | 1152 C1 C1 C1 C1 C1 C1 C1 \ C1 / \ C1 C1 C1 / \ C1 C1 C1 C1 C1 / \ C1 / \ C1 /|
x *|-2 - ---| x *|8 + --- + --------| x *|-48 - --- + ---------------------------------------- - ------------ + ------------| x *|384 + ---- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------- - -------------------------------------------- + ---------------------------------------------------------------------------- + ------------ + -------------|
| 2| | 2 2 | | 2 2 2 2 | | 2 2 2 2 2 2 2 |
x \ C1 / \ C1 C1 / \ C1 C1 C1 C1 / \ C1 C1 C1 C1 C1 C1 C1 / / 6\
y(x) = C1 + --- + ------------- + ----------------------- + --------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + O\x /
3 5 7 9 11
C1 2*C1 6*C1 24*C1 120*C1
$$y{\left(x \right)} = \frac{x^{5} \left(384 - \frac{32 \left(-7 - \frac{18}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{8 \left(10 + \frac{21}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{24 \left(18 + \frac{55}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{2 \left(-56 + \frac{12 \left(-5 - \frac{14}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{3 \left(10 + \frac{21}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{144}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{2 \left(216 - \frac{12 \left(-7 - \frac{18}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{12 \left(-5 - \frac{14}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{5 \left(10 + \frac{21}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{16 \left(14 + \frac{45}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{660}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{432 - \frac{24 \left(-7 - \frac{18}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{24 \left(-5 - \frac{14}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{10 \left(10 + \frac{21}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{32 \left(14 + \frac{45}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{3 \left(-56 + \frac{12 \left(-5 - \frac{14}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{3 \left(10 + \frac{21}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{144}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{4 \left(112 - \frac{18 \left(-5 - \frac{14}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{21 \left(5 + \frac{18}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{3 \left(10 + \frac{21}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{360}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{1320}{C_{1}^{2}}}{C_{1}^{2}} + \frac{1152}{C_{1}^{2}}\right)}{120 C_{1}^{11}} + \frac{x^{4} \left(-48 + \frac{8 \left(-7 - \frac{18}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{2 \left(10 + \frac{21}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{-56 + \frac{12 \left(-5 - \frac{14}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{3 \left(10 + \frac{21}{C_{1}^{2}}\right)}{C_{1}^{2}} - \frac{144}{C_{1}^{2}}}{C_{1}^{2}} - \frac{120}{C_{1}^{2}}\right)}{24 C_{1}^{9}} + \frac{x^{3} \left(8 + \frac{10 + \frac{21}{C_{1}^{2}}}{C_{1}^{2}} + \frac{16}{C_{1}^{2}}\right)}{6 C_{1}^{7}} + \frac{x^{2} \left(-2 - \frac{3}{C_{1}^{2}}\right)}{2 C_{1}^{5}} + \frac{x}{C_{1}^{3}} + C_{1} + 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
[src]
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.5521911348498392)
(-5.555555555555555, -3.840129630073383e-08)
(-3.333333333333333, nan)
(-1.1111111111111107, 2.78363573e-315)
(1.1111111111111107, 8.427456047434801e+197)
(3.333333333333334, 3.1933833808213433e-248)
(5.555555555555557, 1.5636038433718505e+185)
(7.777777777777779, 8.388243567719907e+296)
(10.0, 3.861029683e-315)
(10.0, 3.861029683e-315)