Sr Examen
Ecuación diferencial xdx+ydy=(xdy-ydx)/(x^(2)+y^(2))
El profesor se sorprenderá mucho al ver tu solución correcta😉
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Solución
Ha introducido
[src]
d dy*x dx*y(x)
x + --(y(x))*y(x) = ---------------- - ----------------
dx 2 2 2 2
dx*x + dx*y (x) dx*x + dx*y (x)
$$x + y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} = - \frac{dx y{\left(x \right)}}{dx x^{2} + dx y^{2}{\left(x \right)}} + \frac{dy x}{dx x^{2} + dx y^{2}{\left(x \right)}}$$
x + y*y' = -dx*y/(dx*x^2 + dx*y^2) + dy*x/(dx*x^2 + dx*y^2)
Respuesta
[src]
/ / 10 6*dy \ / 15*dy \ \ / / 10 6*dy \ / 15*dy \ / 90 42*dy \ / 21*dy \ / 10 3*dy \ / 3*dy \\
| 8*|1 + --- - ------| 2*|2 - ------| | | 8*|1 + --- - ------| 4*|2 - ------| 4*|-5 - --- + ------| 10*|-2 + ------| 2*dy*|1 + --- - ------| 6*dy*|1 - ------||
| | 4 2 | | 2 | | | | 4 2 | | 2 | | 4 2 | | 2 | | 4 2 | | 2 ||
2 / 2 dy \ 3 / 5 4*dy \ 4 | 12 / dy \ / 10 3*dy \ \ C1 C1 *dx/ \ C1 *dx/ / 3*dy \ / dy \ 6*dy | 5 | 136 / dy \ / 10 3*dy \ \ C1 C1 *dx/ / dy \ / 6*dy \ \ C1 *dx/ / dy \ / 30 6*dy \ / dy \ / 15*dy \ / 3*dy \ / dy \ \ C1 C1 *dx/ / dy \ / 10 6*dy \ \ C1 *dx/ 90*dy \ C1 C1 *dx/ \ C1 *dx/|
x *|-1 - --- + ------| x *|-1 - --- + ------| x *|--- + |-1 + ------|*|1 + --- - ------| - -------------------- - -------------- + 2*|1 - ------|*|-1 + ------| - ------| x *|4 + --- + |-1 + ------|*|1 + --- - ------| - -------------------- - 6*|-1 + ------|*|-1 + ------| - -------------- - 2*|-1 + ------|*|-1 - --- + ------| + 2*|-1 + ------|*|2 - ------| + 3*|1 - ------|*|-1 + ------| + --------------------- + 4*|-1 + ------|*|1 + --- - ------| + ---------------- - ------ + ----------------------- + -----------------|
| 4 2 | | 4 2 | | 4 | 2 | | 4 2 | 4 4 | 2 | | 2 | 2 | | 4 | 2 | | 4 2 | 4 | 2 | | 2 | 4 | 2 | | 4 2 | | 2 | | 2 | | 2 | | 2 | 4 | 2 | | 4 2 | 4 2 2 2 |
x \ C1 C1 *dx/ \ C1 C1 *dx/ \C1 \ C1 *dx/ \ C1 C1 *dx/ C1 C1 \ C1 *dx/ \ C1 *dx/ C1 *dx/ \ C1 \ C1 *dx/ \ C1 C1 *dx/ C1 \ C1 *dx/ \ C1 *dx/ C1 \ C1 *dx/ \ C1 C1 *dx/ \ C1 *dx/ \ C1 *dx/ \ C1 *dx/ \ C1 *dx/ C1 \ C1 *dx/ \ C1 C1 *dx/ C1 C1 *dx C1 *dx C1 *dx / / 6\
y(x) = C1 - --- + ---------------------- + ---------------------- + --------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + O\x /
2 2*C1 4 3 6
C1 3*C1 24*C1 60*C1
$$y{\left(x \right)} = \frac{x^{5} \left(- 6 \left(-1 + \frac{dy}{C_{1}^{2} dx}\right) \left(-1 + \frac{6 dy}{C_{1}^{2} dx}\right) + 3 \left(-1 + \frac{dy}{C_{1}^{2} dx}\right) \left(1 - \frac{3 dy}{C_{1}^{2} dx}\right) + 2 \left(-1 + \frac{dy}{C_{1}^{2} dx}\right) \left(2 - \frac{15 dy}{C_{1}^{2} dx}\right) - 2 \left(-1 + \frac{dy}{C_{1}^{2} dx}\right) \left(-1 + \frac{6 dy}{C_{1}^{2} dx} - \frac{30}{C_{1}^{4}}\right) + 4 \left(-1 + \frac{dy}{C_{1}^{2} dx}\right) \left(1 - \frac{6 dy}{C_{1}^{2} dx} + \frac{10}{C_{1}^{4}}\right) + \left(-1 + \frac{dy}{C_{1}^{2} dx}\right) \left(1 - \frac{3 dy}{C_{1}^{2} dx} + \frac{10}{C_{1}^{4}}\right) + 4 + \frac{6 dy \left(1 - \frac{3 dy}{C_{1}^{2} dx}\right)}{C_{1}^{2} dx} + \frac{2 dy \left(1 - \frac{3 dy}{C_{1}^{2} dx} + \frac{10}{C_{1}^{4}}\right)}{C_{1}^{2} dx} - \frac{90 dy}{C_{1}^{2} dx} + \frac{10 \left(-2 + \frac{21 dy}{C_{1}^{2} dx}\right)}{C_{1}^{4}} - \frac{4 \left(2 - \frac{15 dy}{C_{1}^{2} dx}\right)}{C_{1}^{4}} + \frac{4 \left(-5 + \frac{42 dy}{C_{1}^{2} dx} - \frac{90}{C_{1}^{4}}\right)}{C_{1}^{4}} - \frac{8 \left(1 - \frac{6 dy}{C_{1}^{2} dx} + \frac{10}{C_{1}^{4}}\right)}{C_{1}^{4}} + \frac{136}{C_{1}^{4}}\right)}{60 C_{1}^{6}} + \frac{x^{3} \left(-1 + \frac{4 dy}{C_{1}^{2} dx} - \frac{5}{C_{1}^{4}}\right)}{3 C_{1}^{4}} + \frac{x^{4} \left(2 \left(-1 + \frac{dy}{C_{1}^{2} dx}\right) \left(1 - \frac{3 dy}{C_{1}^{2} dx}\right) + \left(-1 + \frac{dy}{C_{1}^{2} dx}\right) \left(1 - \frac{3 dy}{C_{1}^{2} dx} + \frac{10}{C_{1}^{4}}\right) - \frac{6 dy}{C_{1}^{2} dx} - \frac{2 \left(2 - \frac{15 dy}{C_{1}^{2} dx}\right)}{C_{1}^{4}} - \frac{8 \left(1 - \frac{6 dy}{C_{1}^{2} dx} + \frac{10}{C_{1}^{4}}\right)}{C_{1}^{4}} + \frac{12}{C_{1}^{4}}\right)}{24 C_{1}^{3}} - \frac{x}{C_{1}^{2}} + \frac{x^{2} \left(-1 + \frac{dy}{C_{1}^{2} dx} - \frac{2}{C_{1}^{4}}\right)}{2 C_{1}} + C_{1} + O\left(x^{6}\right)$$
Clasificación
1st power series
lie group