Sr Examen
Ecuación diferencial y*(x^2)*(dy)-x*(y^2)*(dx)=2*(dx)-(y*(dy))
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
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2 2 d d
- x*y (x) + x *--(y(x))*y(x) = 2 - --(y(x))*y(x)
dx dx
$$x^{2} y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} - x y^{2}{\left(x \right)} = - y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 2$$
x^2*y*y' - x*y^2 = -y*y' + 2
Respuesta
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____________________________________________________________________________________
/ 2 2 2
y(x) = -\/ C1 + 2*x + I*log(I + x) + C1*x - I*log(x - I) + I*x *log(I + x) - I*x *log(x - I)
$$y{\left(x \right)} = - \sqrt{C_{1} x^{2} + C_{1} - i x^{2} \log{\left(x - i \right)} + i x^{2} \log{\left(x + i \right)} + 2 x - i \log{\left(x - i \right)} + i \log{\left(x + i \right)}}$$
____________________________________________________________________________________
/ 2 2 2
y(x) = \/ C1 + 2*x + I*log(I + x) + C1*x - I*log(x - I) + I*x *log(I + x) - I*x *log(x - I)
$$y{\left(x \right)} = \sqrt{C_{1} x^{2} + C_{1} - i x^{2} \log{\left(x - i \right)} + i x^{2} \log{\left(x + i \right)} + 2 x - i \log{\left(x - i \right)} + i \log{\left(x + i \right)}}$$
Gráfico para el problema de Cauchy
Clasificación
1st exact
Bernoulli
1st power series
lie group
1st exact Integral
Bernoulli Integral
Respuesta numérica
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(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.6575146677780436)
(-5.555555555555555, 0.6115115791072541)
(-3.333333333333333, 0.6667524357667878)
(-1.1111111111111107, 1.0306935301610332)
(1.1111111111111107, 3.6051181944457986)
(3.333333333333334, 8.703722891991598)
(5.555555555555557, 14.145871684132752)
(7.777777777777779, 19.658782425820153)
(10.0, 25.197184607474927)
(10.0, 25.197184607474927)