Sr Examen
Ecuación diferencial 2xydy/dxdx+(2y^2)=3x−6
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Solución
Ha introducido
[src]
2 2*y (x) d ------- + 2*x*--(y(x))*y(x) = -6 + 3*x dx dx
$$2 x y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + \frac{2 y^{2}{\left(x \right)}}{dx} = 3 x - 6$$
2*x*y*y' + 2*y^2/dx = 3*x - 6
Respuesta
[src]
/ ________________________________________________________________________________________________________________________________________________
| / 2*log(x) 2*log(x) 2*log(x)
| / -------- -------- --------
| / dx 2 dx dx
| / 2*C1 C1*dx 6*dx*e 3*dx *e 3*dx*x*e
|- / -------------------------- + -------------------------- - -------------------------- - -------------------------- + -------------------------- for dx != -2
| / 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) = < \/ 2*e + dx*e 2*e + dx*e 2*e + dx*e 2*e + dx*e 2*e + dx*e
|
| ____________________________________________________
| / -2*log(x)
| / -2*log(x) -2*log(x) ---------
| / --------- --------- dx
| / dx dx 6*e
| - / C1*e + 3*e *log(x) + ------------ otherwise
| \/ x
\
$$y{\left(x \right)} = \begin{cases} - \sqrt{\frac{C_{1} dx}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{2 C_{1}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}} - \frac{3 dx^{2} e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{3 dx x e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}} - \frac{6 dx e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}}} & \text{for}\: dx \neq -2 \\- \sqrt{C_{1} e^{- \frac{2 \log{\left(x \right)}}{dx}} + 3 e^{- \frac{2 \log{\left(x \right)}}{dx}} \log{\left(x \right)} + \frac{6 e^{- \frac{2 \log{\left(x \right)}}{dx}}}{x}} & \text{otherwise} \end{cases}$$
/ ________________________________________________________________________________________________________________________________________________
| / 2*log(x) 2*log(x) 2*log(x)
| / -------- -------- --------
| / dx 2 dx dx
| / 2*C1 C1*dx 6*dx*e 3*dx *e 3*dx*x*e
| / -------------------------- + -------------------------- - -------------------------- - -------------------------- + -------------------------- for dx != -2
| / 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) = <\/ 2*e + dx*e 2*e + dx*e 2*e + dx*e 2*e + dx*e 2*e + dx*e
|
| ____________________________________________________
| / -2*log(x)
| / -2*log(x) -2*log(x) ---------
| / --------- --------- dx
| / dx dx 6*e
| / C1*e + 3*e *log(x) + ------------ otherwise
| \/ x
\
$$y{\left(x \right)} = \begin{cases} \sqrt{\frac{C_{1} dx}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{2 C_{1}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}} - \frac{3 dx^{2} e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}} + \frac{3 dx x e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}} - \frac{6 dx e^{\frac{2 \log{\left(x \right)}}{dx}}}{dx e^{\frac{2 \log{\left(x \right)}}{dx}} + 2 e^{\frac{2 \log{\left(x \right)}}{dx}}}} & \text{for}\: dx \neq -2 \\\sqrt{C_{1} e^{- \frac{2 \log{\left(x \right)}}{dx}} + 3 e^{- \frac{2 \log{\left(x \right)}}{dx}} \log{\left(x \right)} + \frac{6 e^{- \frac{2 \log{\left(x \right)}}{dx}}}{x}} & \text{otherwise} \end{cases}$$
Clasificación
Bernoulli
almost linear
lie group
Bernoulli Integral
almost linear Integral