Sr Examen
Ecuación diferencial dydx=x^2/y^2+1+y/x
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Solución
Ha introducido
[src]
2
y(x) x
dy = 1 + ---- + -----
x 2
y (x)
$$dy = \frac{x^{2}}{y^{2}{\left(x \right)}} + 1 + \frac{y{\left(x \right)}}{x}$$
dy = x^2/y^2 + 1 + y/x
Respuesta
[src]
___________________________________________________________________________
/ ___________________________________________
/ / / 2 \
2/3 3 / / 6 |/ 3\ 6| 3 3 3 3 ___ 2 2
x 2 *\/ \/ x *\\-27 + 2*(-1 + dy) / - 4*(-1 + dy) / + 27*x - 2*x *(-1 + dy) dy*x \/ 2 *x *(-1 + dy)
y(x) = - - - ------------------------------------------------------------------------------------- + ---- - ----------------------------------------------------------------------------------
3 6 3 ___________________________________________________________________________
/ ___________________________________________
/ / / 2 \
3 / / 6 |/ 3\ 6| 3 3 3
3*\/ \/ x *\\-27 + 2*(-1 + dy) / - 4*(-1 + dy) / + 27*x - 2*x *(-1 + dy)
$$y{\left(x \right)} = \frac{dy x}{3} - \frac{\sqrt[3]{2} x^{2} \left(dy - 1\right)^{2}}{3 \sqrt[3]{- 2 x^{3} \left(dy - 1\right)^{3} + 27 x^{3} + \sqrt{x^{6} \left(- 4 \left(dy - 1\right)^{6} + \left(2 \left(dy - 1\right)^{3} - 27\right)^{2}\right)}}} - \frac{x}{3} - \frac{2^{\frac{2}{3}} \sqrt[3]{- 2 x^{3} \left(dy - 1\right)^{3} + 27 x^{3} + \sqrt{x^{6} \left(- 4 \left(dy - 1\right)^{6} + \left(2 \left(dy - 1\right)^{3} - 27\right)^{2}\right)}}}{6}$$
__________________________________________________________________________ / __________________________________________________________________________ \
/ __________________________________________ | / __________________________________________ |
/ / / 2 \ | / / / 2 \ |
3 / / 6 |/ 3\ 6| 3 3 3 / ___\ | 2/3 3 / / 6 |/ 3\ 6| 3 3 3 / ___\| 3 ___ 2 2
\/ \/ x *\\27 - 2*(-1 + dy) / - 4*(-1 + dy) / + 27*x - 2*x *(-1 + dy) *\1 + I*\/ 3 /*\-4*x + 4*dy*x + 2 *\/ \/ x *\\27 - 2*(-1 + dy) / - 4*(-1 + dy) / + 27*x - 2*x *(-1 + dy) *\1 + I*\/ 3 // + 8*\/ 2 *x *(-1 + dy)
y(x) = ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
__________________________________________________________________________
/ __________________________________________
/ / / 2 \
/ ___\ 3 / / 6 |/ 3\ 6| 3 3 3
12*\1 + I*\/ 3 /*\/ \/ x *\\27 - 2*(-1 + dy) / - 4*(-1 + dy) / + 27*x - 2*x *(-1 + dy)
$$y{\left(x \right)} = \frac{8 \sqrt[3]{2} x^{2} \left(dy - 1\right)^{2} + \left(1 + \sqrt{3} i\right) \left(4 dy x - 4 x + 2^{\frac{2}{3}} \left(1 + \sqrt{3} i\right) \sqrt[3]{- 2 x^{3} \left(dy - 1\right)^{3} + 27 x^{3} + \sqrt{x^{6} \left(\left(27 - 2 \left(dy - 1\right)^{3}\right)^{2} - 4 \left(dy - 1\right)^{6}\right)}}\right) \sqrt[3]{- 2 x^{3} \left(dy - 1\right)^{3} + 27 x^{3} + \sqrt{x^{6} \left(\left(27 - 2 \left(dy - 1\right)^{3}\right)^{2} - 4 \left(dy - 1\right)^{6}\right)}}}{12 \left(1 + \sqrt{3} i\right) \sqrt[3]{- 2 x^{3} \left(dy - 1\right)^{3} + 27 x^{3} + \sqrt{x^{6} \left(\left(27 - 2 \left(dy - 1\right)^{3}\right)^{2} - 4 \left(dy - 1\right)^{6}\right)}}}$$
__________________________________________________________________________ / __________________________________________________________________________ \
/ __________________________________________ | / __________________________________________ |
/ / / 2 \ | / / / 2 \ |
3 / / 6 |/ 3\ 6| 3 3 3 / ___\ | 2/3 3 / / 6 |/ 3\ 6| 3 3 3 / ___\| 3 ___ 2 2
\/ \/ x *\\27 - 2*(-1 + dy) / - 4*(-1 + dy) / + 27*x - 2*x *(-1 + dy) *\1 - I*\/ 3 /*\-4*x + 4*dy*x + 2 *\/ \/ x *\\27 - 2*(-1 + dy) / - 4*(-1 + dy) / + 27*x - 2*x *(-1 + dy) *\1 - I*\/ 3 // + 8*\/ 2 *x *(-1 + dy)
y(x) = ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
__________________________________________________________________________
/ __________________________________________
/ / / 2 \
/ ___\ 3 / / 6 |/ 3\ 6| 3 3 3
12*\1 - I*\/ 3 /*\/ \/ x *\\27 - 2*(-1 + dy) / - 4*(-1 + dy) / + 27*x - 2*x *(-1 + dy)
$$y{\left(x \right)} = \frac{8 \sqrt[3]{2} x^{2} \left(dy - 1\right)^{2} + \left(1 - \sqrt{3} i\right) \left(4 dy x - 4 x + 2^{\frac{2}{3}} \left(1 - \sqrt{3} i\right) \sqrt[3]{- 2 x^{3} \left(dy - 1\right)^{3} + 27 x^{3} + \sqrt{x^{6} \left(\left(27 - 2 \left(dy - 1\right)^{3}\right)^{2} - 4 \left(dy - 1\right)^{6}\right)}}\right) \sqrt[3]{- 2 x^{3} \left(dy - 1\right)^{3} + 27 x^{3} + \sqrt{x^{6} \left(\left(27 - 2 \left(dy - 1\right)^{3}\right)^{2} - 4 \left(dy - 1\right)^{6}\right)}}}{12 \left(1 - \sqrt{3} i\right) \sqrt[3]{- 2 x^{3} \left(dy - 1\right)^{3} + 27 x^{3} + \sqrt{x^{6} \left(\left(27 - 2 \left(dy - 1\right)^{3}\right)^{2} - 4 \left(dy - 1\right)^{6}\right)}}}$$
Clasificación
nth algebraic
nth algebraic Integral