Sr Examen
Ecuación diferencial xydy-1+y^2/1+x^2dx=0
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Solución
Ha introducido
[src]
2
2 1 y (x) d
x - -- + ----- + x*--(y(x))*y(x) = 0
dx dx dx
$$x^{2} + x y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + \frac{y^{2}{\left(x \right)}}{dx} - \frac{1}{dx} = 0$$
x^2 + x*y*y' + y^2/dx - 1/dx = 0
Respuesta
[src]
/ ______________________________________________________________________________________________________________________________________
| / 2*log(x) 2*log(x) 2*log(x)
| / -------- -------- --------
| / dx dx 2 dx
| / C1 e C1*dx dx*e dx*x *e
|- / ------------------------ + ------------------------ + ------------------------ + ------------------------ - ------------------------ for dx != -1
| / 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x)
| / -------- -------- -------- -------- -------- -------- -------- -------- -------- --------
| / dx dx dx dx dx dx dx dx dx dx
y(x) = < \/ dx*e + e dx*e + e dx*e + e dx*e + e dx*e + e
|
| __________________________________________________
| / -2*log(x) -2*log(x)
| / -2*log(x) --------- ---------
| / --------- dx dx
| / dx e 2*e *log(x)
| - / C1*e - ---------- + ------------------- otherwise
| / 2 dx
\ \/ dx*x
$$y{\left(x \right)} = \begin{cases} - \sqrt{\frac{C_{1} dx}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{C_{1}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}} - \frac{dx x^{2} e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{dx e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}}} & \text{for}\: dx \neq -1 \\- \sqrt{C_{1} e^{- \frac{2 \log{\left(x \right)}}{dx}} + \frac{2 e^{- \frac{2 \log{\left(x \right)}}{dx}} \log{\left(x \right)}}{dx} - \frac{e^{- \frac{2 \log{\left(x \right)}}{dx}}}{dx x^{2}}} & \text{otherwise} \end{cases}$$
/ ______________________________________________________________________________________________________________________________________
| / 2*log(x) 2*log(x) 2*log(x)
| / -------- -------- --------
| / dx dx 2 dx
| / C1 e C1*dx dx*e dx*x *e
| / ------------------------ + ------------------------ + ------------------------ + ------------------------ - ------------------------ for dx != -1
| / 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x) 2*log(x)
| / -------- -------- -------- -------- -------- -------- -------- -------- -------- --------
| / dx dx dx dx dx dx dx dx dx dx
y(x) = <\/ dx*e + e dx*e + e dx*e + e dx*e + e dx*e + e
|
| __________________________________________________
| / -2*log(x) -2*log(x)
| / -2*log(x) --------- ---------
| / --------- dx dx
| / dx e 2*e *log(x)
| / C1*e - ---------- + ------------------- otherwise
| / 2 dx
\ \/ dx*x
$$y{\left(x \right)} = \begin{cases} \sqrt{\frac{C_{1} dx}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{C_{1}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}} - \frac{dx x^{2} e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{dx e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + e^{\frac{2 \log{\left(x \right)}}{dx}}}} & \text{for}\: dx \neq -1 \\\sqrt{C_{1} e^{- \frac{2 \log{\left(x \right)}}{dx}} + \frac{2 e^{- \frac{2 \log{\left(x \right)}}{dx}} \log{\left(x \right)}}{dx} - \frac{e^{- \frac{2 \log{\left(x \right)}}{dx}}}{dx x^{2}}} & \text{otherwise} \end{cases}$$
Clasificación
almost linear
lie group
almost linear Integral