Sr Examen
Ecuación diferencial cosydx+(3-xsiny)dy=0
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
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d d 3*--(y(x)) - x*--(y(x))*sin(y(x)) + cos(y(x)) = 0 dx dx
$$- x \sin{\left(y{\left(x \right)} \right)} \frac{d}{d x} y{\left(x \right)} + \cos{\left(y{\left(x \right)} \right)} + 3 \frac{d}{d x} y{\left(x \right)} = 0$$
-x*sin(y)*y' + cos(y) + 3*y' = 0
Respuesta
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2 3 / 2 2 \ 5 / 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 2 \ 2 2 \ 4 / 2 2 \
x*cos(C1) x *cos(C1)*sin(C1) x *\cos (C1) - 2*sin (C1)/*cos(C1) x *\- 54*sin (C1) - 20*sin (C1)*\sin (C1) - cos (C1)/ - 12*sin (C1)*\sin (C1) - 2*cos (C1)/ - 6*sin (C1)*\sin (C1) - 3*cos (C1)/ - 6*sin (C1)*\- 7*cos (C1) + 2*sin (C1)/ - 4*sin (C1)*\- 11*cos (C1) + 4*sin (C1)/ + 2*cos (C1)*\- 11*cos (C1) + 34*sin (C1)/ + 3*cos (C1)*\- 7*cos (C1) + 20*sin (C1)/ + 6*cos (C1)*\- 2*cos (C1) + 7*sin (C1)/ + 10*cos (C1)*\sin (C1) - cos (C1)/ + 112*cos (C1)*sin (C1)/*cos(C1) x *\- 3*sin (C1) + 5*cos (C1)/*cos(C1)*sin(C1) / 6\
y(x) = C1 - --------- - ------------------ + ---------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------- + O\x /
3 9 54 29160 243
$$y{\left(x \right)} = - \frac{x \cos{\left(C_{1} \right)}}{3} - \frac{x^{2} \sin{\left(C_{1} \right)} \cos{\left(C_{1} \right)}}{9} + \frac{x^{3} \left(- 2 \sin^{2}{\left(C_{1} \right)} + \cos^{2}{\left(C_{1} \right)}\right) \cos{\left(C_{1} \right)}}{54} + \frac{x^{4} \left(- 3 \sin^{2}{\left(C_{1} \right)} + 5 \cos^{2}{\left(C_{1} \right)}\right) \sin{\left(C_{1} \right)} \cos{\left(C_{1} \right)}}{243} + \frac{x^{5} \left(- 6 \left(\sin^{2}{\left(C_{1} \right)} - 3 \cos^{2}{\left(C_{1} \right)}\right) \sin^{2}{\left(C_{1} \right)} - 12 \left(\sin^{2}{\left(C_{1} \right)} - 2 \cos^{2}{\left(C_{1} \right)}\right) \sin^{2}{\left(C_{1} \right)} - 20 \left(\sin^{2}{\left(C_{1} \right)} - \cos^{2}{\left(C_{1} \right)}\right) \sin^{2}{\left(C_{1} \right)} + 10 \left(\sin^{2}{\left(C_{1} \right)} - \cos^{2}{\left(C_{1} \right)}\right) \cos^{2}{\left(C_{1} \right)} - 6 \left(2 \sin^{2}{\left(C_{1} \right)} - 7 \cos^{2}{\left(C_{1} \right)}\right) \sin^{2}{\left(C_{1} \right)} - 4 \left(4 \sin^{2}{\left(C_{1} \right)} - 11 \cos^{2}{\left(C_{1} \right)}\right) \sin^{2}{\left(C_{1} \right)} + 6 \left(7 \sin^{2}{\left(C_{1} \right)} - 2 \cos^{2}{\left(C_{1} \right)}\right) \cos^{2}{\left(C_{1} \right)} + 3 \left(20 \sin^{2}{\left(C_{1} \right)} - 7 \cos^{2}{\left(C_{1} \right)}\right) \cos^{2}{\left(C_{1} \right)} + 2 \left(34 \sin^{2}{\left(C_{1} \right)} - 11 \cos^{2}{\left(C_{1} \right)}\right) \cos^{2}{\left(C_{1} \right)} - 54 \sin^{4}{\left(C_{1} \right)} + 112 \sin^{2}{\left(C_{1} \right)} \cos^{2}{\left(C_{1} \right)}\right) \cos{\left(C_{1} \right)}}{29160} + C_{1} + O\left(x^{6}\right)$$
Gráfico para el problema de Cauchy
Clasificación
factorable
1st exact
1st power series
lie group
1st exact Integral
Respuesta numérica
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(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.5377369352062176)
(-5.555555555555555, 0.14378000955366288)
(-3.333333333333333, -0.907396066474991)
(-1.1111111111111107, 6.91571707653673e-310)
(1.1111111111111107, 4.6409274941778e-310)
(3.333333333333334, 7.933388671689103e-100)
(5.555555555555557, 6.91571996333146e-310)
(7.777777777777779, 6.9157159477224e-310)
(10.0, -2.237897020109815e-74)
(10.0, -2.237897020109815e-74)