Sr Examen
Ecuación diferencial dx*(x*y^3+x)+dy*(x^2*y^2+y^2)=0
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
[src]
3 2 d 2 2 d
x + x*y (x) + y (x)*--(y(x)) + x *y (x)*--(y(x)) = 0
dx dx
$$x^{2} y^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + x y^{3}{\left(x \right)} + x + y^{2}{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = 0$$
x^2*y^2*y' + x*y^3 + x + y^2*y' = 0
Respuesta
[src]
_________________
/ C1 / 3 ____ 5/6 ___\
/ 1 + ----------- *\- \/ -1 + (-1) *\/ 3 /
/ 3/2
3 / / 2\
\/ \1 + x /
y(x) = --------------------------------------------------
2
$$y{\left(x \right)} = \frac{\left(- \sqrt[3]{-1} + \left(-1\right)^{\frac{5}{6}} \sqrt{3}\right) \sqrt[3]{\frac{C_{1}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + 1}}{2}$$
_________________
/ C1 / 3 ____ 5/6 ___\
/ 1 + ----------- *\- \/ -1 - (-1) *\/ 3 /
/ 3/2
3 / / 2\
\/ \1 + x /
y(x) = --------------------------------------------------
2
$$y{\left(x \right)} = \frac{\left(- \sqrt[3]{-1} - \left(-1\right)^{\frac{5}{6}} \sqrt{3}\right) \sqrt[3]{\frac{C_{1}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} + 1}}{2}$$
__________________
/ C1
y(x) = / -1 + -----------
/ 3/2
3 / / 2\
\/ \1 + x /
$$y{\left(x \right)} = \sqrt[3]{\frac{C_{1}}{\left(x^{2} + 1\right)^{\frac{3}{2}}} - 1}$$
Gráfico para el problema de Cauchy
Clasificación
separable
1st exact
almost linear
1st power series
lie group
separable Integral
1st exact Integral
almost linear Integral
Respuesta numérica
[src]
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 1.2584340779835772)
(-5.555555555555555, 1.9151116685706784)
(-3.333333333333333, 3.2153777341063625)
(-1.1111111111111107, 7.554092834166894)
(1.1111111111111107, 7.554093052806683)
(3.333333333333334, 3.2153774249623193)
(5.555555555555557, 1.9151117379327802)
(7.777777777777779, 1.2584342386361878)
(10.0, 0.7499999263712305)
(10.0, 0.7499999263712305)