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Ecuaciones diferenciales/ xydy-1+y^2/1+x^2dx=0

Ecuación diferencial xydy-1+y^2/1+x^2dx=0

El profesor se sorprenderá mucho al ver tu solución correcta😉

v

Para el problema de Cauchy:

y() =
y'() =
y''() =
y'''() =
y''''() =

Gráfico:

interior superior
⌨

Solución

Ha introducido [src] 
           2                         
 2   1    y (x)     d                
x  - -- + ----- + x*--(y(x))*y(x) = 0
     dx     dx      dx
x2+xy(x)ddxy(x)+y2(x)dx−1dx=0x^{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(x)={−C1dxdxe2log⁡(x)dx+e2log⁡(x)dx+C1dxe2log⁡(x)dx+e2log⁡(x)dx−dxx2e2log⁡(x)dxdxe2log⁡(x)dx+e2log⁡(x)dx+dxe2log⁡(x)dxdxe2log⁡(x)dx+e2log⁡(x)dx+e2log⁡(x)dxdxe2log⁡(x)dx+e2log⁡(x)dxfor dx≠−1−C1e−2log⁡(x)dx+2e−2log⁡(x)dxlog⁡(x)dx−e−2log⁡(x)dxdxx2otherwisey{\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}
Trazar el gráfico con C=-1
Trazar el gráfico con C=0
Trazar el gráfico con C=1 
       /          ______________________________________________________________________________________________________________________________________              
       |         /                                    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(x)={C1dxdxe2log⁡(x)dx+e2log⁡(x)dx+C1dxe2log⁡(x)dx+e2log⁡(x)dx−dxx2e2log⁡(x)dxdxe2log⁡(x)dx+e2log⁡(x)dx+dxe2log⁡(x)dxdxe2log⁡(x)dx+e2log⁡(x)dx+e2log⁡(x)dxdxe2log⁡(x)dx+e2log⁡(x)dxfor dx≠−1C1e−2log⁡(x)dx+2e−2log⁡(x)dxlog⁡(x)dx−e−2log⁡(x)dxdxx2otherwisey{\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}
Trazar el gráfico con C=-1
Trazar el gráfico con C=0
Trazar el gráfico con C=1
Clasificación
almost linear
lie group
almost linear Integral
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