Sr Examen
Ecuación diferencial (y^2e^2x+1)dx+(2ye^2x-e^x)dy=0
El profesor se sorprenderá mucho al ver tu solución correcta😉
⌨
Solución
Ha introducido
[src]
d x 2 2 d 2
1 - --(y(x))*e + x*y (x)*e + 2*x*--(y(x))*e *y(x) = 0
dx dx
$$x y^{2}{\left(x \right)} e^{2} + 2 x y{\left(x \right)} e^{2} \frac{d}{d x} y{\left(x \right)} - e^{x} \frac{d}{d x} y{\left(x \right)} + 1 = 0$$
x*y^2*exp(2) + 2*x*y*exp(2)*y' - exp(x)*y' + 1 = 0
Respuesta
[src]
/ 2 \ / 3 2 \ / 2 4 2 3 2 2 \
2 / 2 2 2\ 3 | / 2\ 2 2 2 / 2\ 2| 4 | / 2\ 2 / 2\ 2 2 2 2 2 4 / 2\ 2 / 2\ 2 2 / 2\ 2 2 4| 5 | 2 / 2\ / 2\ 4 2 / 2\ 2 2 2 2 4 4 4 / 2\ 2 2 / 2\ 4 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\ 2\ 4 2 / / 2\ 2 2 2\ 4 2 / 2 2 / 2\ 2\ 4 / 2\ 2 2 / 2\ 2 4 / 2\ 2 2 / 2\ 4|
x *\-1 + C1 *e + 2*C1*e / x *\-1 + 2*\1 - 2*C1*e / + 4*e + 4*C1*e - 2*C1 *\1 - 2*C1*e /*e / x *\5 - 6*\1 - 2*C1*e / + 6*e - 24*\1 - 2*C1*e /*e - 12*C1*e - 3*C1 *e + 12*C1 *e - 12*C1*\1 - 2*C1*e /*e - 6*(1 + C1)*\1 - 2*C1*e /*e + 6*C1 *\1 - 2*C1*e / *e + 6*C1 *(1 + C1)*e / x *\13 - 56*e - 36*\1 - 2*C1*e / + 24*\1 - 2*C1*e / + 96*e - 48*C1*e - 40*\1 - 2*C1*e /*e - 4*C1 *e + 16*C1 *e + 32*(1 + C1)*e + 96*C1*e + 144*\1 - 2*C1*e / *e - 96*C1 *\1 - 2*C1*e /*e - 24*C1 *\1 - 2*C1*e / *e - 12*\1 - 2*C1*e /*\-2 + C1 *e - C1*\1 - 2*C1*e / + 4*C1*e /*e - 4*\1 - 2*C1*e /*\-1 + C1 *e - C1*\1 - 2*C1*e / + 4*C1*e /*e - 4*\1 - 2*C1*e /*\-3 - 2*C1*\1 - 2*C1*e / + 2*C1 *e + 8*C1*e /*e + 4*C1 *\-1 + C1 *e - C1*\1 - 2*C1*e / + 4*C1*e /*e + 4*C1 *\-3 - 2*C1*\1 - 2*C1*e / + 2*C1 *e + 8*C1*e /*e + 12*C1 *\-2 + C1 *e - C1*\1 - 2*C1*e / + 4*C1*e /*e + 16*\1 - 2*C1*e / *(1 + C1)*e + 24*C1 *\1 - 2*C1*e /*e + 32*C1*(1 + C1)*e + 48*C1*\1 - 2*C1*e / *e - 16*C1 *(1 + C1)*\1 - 2*C1*e /*e / / 6\
y(x) = C1 + x + -------------------------- + -------------------------------------------------------------------- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + O\x /
2 6 24 120
$$y{\left(x \right)} = x + \frac{x^{2} \left(C_{1}^{2} e^{2} + 2 C_{1} e^{2} - 1\right)}{2} + \frac{x^{3} \left(- 2 C_{1}^{2} \left(- 2 C_{1} e^{2} + 1\right) e^{2} + 4 C_{1} e^{2} + 2 \left(- 2 C_{1} e^{2} + 1\right)^{2} - 1 + 4 e^{2}\right)}{6} + \frac{x^{4} \left(6 C_{1}^{2} \left(C_{1} + 1\right) e^{4} + 6 C_{1}^{2} \left(- 2 C_{1} e^{2} + 1\right)^{2} e^{2} - 3 C_{1}^{2} e^{2} + 12 C_{1}^{2} e^{4} - 12 C_{1} \left(- 2 C_{1} e^{2} + 1\right) e^{2} - 12 C_{1} e^{2} - 6 \left(C_{1} + 1\right) \left(- 2 C_{1} e^{2} + 1\right) e^{2} - 6 \left(- 2 C_{1} e^{2} + 1\right)^{3} - 24 \left(- 2 C_{1} e^{2} + 1\right) e^{2} + 5 + 6 e^{2}\right)}{24} + \frac{x^{5} \left(- 16 C_{1}^{2} \left(C_{1} + 1\right) \left(- 2 C_{1} e^{2} + 1\right) e^{4} - 24 C_{1}^{2} \left(- 2 C_{1} e^{2} + 1\right)^{3} e^{2} - 96 C_{1}^{2} \left(- 2 C_{1} e^{2} + 1\right) e^{4} + 24 C_{1}^{2} \left(- 2 C_{1} e^{2} + 1\right) e^{2} + 12 C_{1}^{2} \left(C_{1}^{2} e^{2} - C_{1} \left(- 2 C_{1} e^{2} + 1\right) + 4 C_{1} e^{2} - 2\right) e^{4} + 4 C_{1}^{2} \left(C_{1}^{2} e^{2} - C_{1} \left(- 2 C_{1} e^{2} + 1\right) + 4 C_{1} e^{2} - 1\right) e^{4} + 4 C_{1}^{2} \left(2 C_{1}^{2} e^{2} - 2 C_{1} \left(- 2 C_{1} e^{2} + 1\right) + 8 C_{1} e^{2} - 3\right) e^{4} - 4 C_{1}^{2} e^{2} + 16 C_{1}^{2} e^{4} + 32 C_{1} \left(C_{1} + 1\right) e^{4} + 48 C_{1} \left(- 2 C_{1} e^{2} + 1\right)^{2} e^{2} - 48 C_{1} e^{2} + 96 C_{1} e^{4} + 16 \left(C_{1} + 1\right) \left(- 2 C_{1} e^{2} + 1\right)^{2} e^{2} + 32 \left(C_{1} + 1\right) e^{4} + 24 \left(- 2 C_{1} e^{2} + 1\right)^{4} - 36 \left(- 2 C_{1} e^{2} + 1\right)^{2} + 144 \left(- 2 C_{1} e^{2} + 1\right)^{2} e^{2} - 12 \left(- 2 C_{1} e^{2} + 1\right) \left(C_{1}^{2} e^{2} - C_{1} \left(- 2 C_{1} e^{2} + 1\right) + 4 C_{1} e^{2} - 2\right) e^{2} - 4 \left(- 2 C_{1} e^{2} + 1\right) \left(C_{1}^{2} e^{2} - C_{1} \left(- 2 C_{1} e^{2} + 1\right) + 4 C_{1} e^{2} - 1\right) e^{2} - 4 \left(- 2 C_{1} e^{2} + 1\right) \left(2 C_{1}^{2} e^{2} - 2 C_{1} \left(- 2 C_{1} e^{2} + 1\right) + 8 C_{1} e^{2} - 3\right) e^{2} - 40 \left(- 2 C_{1} e^{2} + 1\right) e^{2} - 56 e^{2} + 13 + 96 e^{4}\right)}{120} + C_{1} + O\left(x^{6}\right)$$
Gráfico para el problema de Cauchy
Clasificación
1st power series
lie group
Respuesta numérica
[src]
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.2742308321771475)
(-5.555555555555555, 0.16583011681367973)
(-3.333333333333333, 0.18226067897410583)
(-1.1111111111111107, 0.2724993968375903)
(1.1111111111111107, 0.8125879953731757)
(3.333333333333334, 3.1933833808213433e-248)
(5.555555555555557, 1.349452126851315e+161)
(7.777777777777779, 8.388243571809574e+296)
(10.0, 3.861029683e-315)
(10.0, 3.861029683e-315)