Sr Examen
Ecuación diferencial dx*(x^2*y/3+2*x*y/3+y^3/3)+dy*(x^2+y^2)=0
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Solución
Ha introducido
[src]
3 2 y (x) 2 d 2 d x *y(x) 2*x*y(x) ----- + x *--(y(x)) + y (x)*--(y(x)) + ------- + -------- = 0 3 dx dx 3 3
$$\frac{x^{2} y{\left(x \right)}}{3} + x^{2} \frac{d}{d x} y{\left(x \right)} + \frac{2 x y{\left(x \right)}}{3} + \frac{y^{3}{\left(x \right)}}{3} + y^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = 0$$
x^2*y/3 + x^2*y' + 2*x*y/3 + y^3/3 + y^2*y' = 0
Respuesta
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/ / 6 \\ / / / 6 \\ /2 8 \ / 6 \\
| 6*|1 + ---|| | | 6*|1 + ---|| 81*|-- - ---| 6*|-1 + ---||
| | 2|| | | | 2|| |27 2| | 2||
4 |252 / 6 \ \ C1 /| 5 |1404 | 684 6 \ C1 /| \ C1 / \ C1 /| 3 / 6 \
2 / 6 \ x *|--- - C1*|-1 + ---| - -----------| x *|---- - C1*|1 - --- + --- + -----------| - ------------- - ------------| C1*x *|1 + ---|
x *|C1 - --| | 3 | 2| C1 | | 3 | 4 2 2 | C1 C1 | | 2|
C1*x \ C1/ \C1 \ C1 / / \C1 \ C1 C1 C1 / / \ C1 / / 6\
y(x) = C1 - ---- + ------------ + -------------------------------------- + --------------------------------------------------------------------------- - --------------- + O\x /
3 18 1944 29160 162
$$y{\left(x \right)} = \frac{x^{2} \left(C_{1} - \frac{6}{C_{1}}\right)}{18} + \frac{x^{4} \left(- C_{1} \left(-1 + \frac{6}{C_{1}^{2}}\right) - \frac{6 \left(1 + \frac{6}{C_{1}^{2}}\right)}{C_{1}} + \frac{252}{C_{1}^{3}}\right)}{1944} + \frac{x^{5} \left(- C_{1} \left(1 + \frac{6 \left(1 + \frac{6}{C_{1}^{2}}\right)}{C_{1}^{2}} + \frac{6}{C_{1}^{2}} - \frac{684}{C_{1}^{4}}\right) - \frac{6 \left(-1 + \frac{6}{C_{1}^{2}}\right)}{C_{1}} - \frac{81 \left(\frac{2}{27} - \frac{8}{C_{1}^{2}}\right)}{C_{1}} + \frac{1404}{C_{1}^{3}}\right)}{29160} + C_{1} - \frac{C_{1} x}{3} - \frac{C_{1} x^{3} \left(1 + \frac{6}{C_{1}^{2}}\right)}{162} + O\left(x^{6}\right)$$
Gráfico para el problema de Cauchy
Clasificación
1st power series
lie group
Respuesta numérica
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(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.42250652543103684)
(-5.555555555555555, 0.2519541452779983)
(-3.333333333333333, 0.1687338489880829)
(-1.1111111111111107, 0.1663482408031554)
(1.1111111111111107, 0.07131982608201175)
(3.333333333333334, 0.01635710645552383)
(5.555555555555557, 0.005547639692318131)
(7.777777777777779, 0.0021134361113643377)
(10.0, 0.0008521672049704501)
(10.0, 0.0008521672049704501)