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sqrty*d*y=3sqrtx*d*x la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
  ___           ___    
\/ y *d*y = 3*\/ x *d*x
ydy=xd3xy d \sqrt{y} = x d 3 \sqrt{x}
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Gráfica
Respuesta rápida [src] 
      2/3            2/3      
y1 = 3   *re(x) + I*3   *im(x)
y1=323re⁡(x)+323iim⁡(x)y_{1} = 3^{\frac{2}{3}} \operatorname{re}{\left(x\right)} + 3^{\frac{2}{3}} i \operatorname{im}{\left(x\right)} 
         /                    2\     /                    2\
         |  /  3 ___      5/6\ |     |  /  3 ___      5/6\ |
         |  |  \/ 3    I*3   | |     |  |  \/ 3    I*3   | |
y2 = I*im|x*|- ----- - ------| | + re|x*|- ----- - ------| |
         \  \    2       2   / /     \  \    2       2   / /
y2=re⁡(x(−332−356i2)2)+iim⁡(x(−332−356i2)2)y_{2} = \operatorname{re}{\left(x \left(- \frac{\sqrt[3]{3}}{2} - \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} + i \operatorname{im}{\left(x \left(- \frac{\sqrt[3]{3}}{2} - \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} 
         /                    2\     /                    2\
         |  /  3 ___      5/6\ |     |  /  3 ___      5/6\ |
         |  |  \/ 3    I*3   | |     |  |  \/ 3    I*3   | |
y3 = I*im|x*|- ----- + ------| | + re|x*|- ----- + ------| |
         \  \    2       2   / /     \  \    2       2   / /
y3=re⁡(x(−332+356i2)2)+iim⁡(x(−332+356i2)2)y_{3} = \operatorname{re}{\left(x \left(- \frac{\sqrt[3]{3}}{2} + \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} + i \operatorname{im}{\left(x \left(- \frac{\sqrt[3]{3}}{2} + \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} 
y3 = re(x*(-3^(1/3)/2 + 3^(5/6)*i/2)^2) + i*im(x*(-3^(1/3)/2 + 3^(5/6)*i/2)^2)
Suma y producto de raíces [src] 
suma
                                /                    2\     /                    2\       /                    2\     /                    2\
                                |  /  3 ___      5/6\ |     |  /  3 ___      5/6\ |       |  /  3 ___      5/6\ |     |  /  3 ___      5/6\ |
 2/3            2/3             |  |  \/ 3    I*3   | |     |  |  \/ 3    I*3   | |       |  |  \/ 3    I*3   | |     |  |  \/ 3    I*3   | |
3   *re(x) + I*3   *im(x) + I*im|x*|- ----- - ------| | + re|x*|- ----- - ------| | + I*im|x*|- ----- + ------| | + re|x*|- ----- + ------| |
                                \  \    2       2   / /     \  \    2       2   / /       \  \    2       2   / /     \  \    2       2   / /
((323re⁡(x)+323iim⁡(x))+(re⁡(x(−332−356i2)2)+iim⁡(x(−332−356i2)2)))+(re⁡(x(−332+356i2)2)+iim⁡(x(−332+356i2)2))\left(\left(3^{\frac{2}{3}} \operatorname{re}{\left(x\right)} + 3^{\frac{2}{3}} i \operatorname{im}{\left(x\right)}\right) + \left(\operatorname{re}{\left(x \left(- \frac{\sqrt[3]{3}}{2} - \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} + i \operatorname{im}{\left(x \left(- \frac{\sqrt[3]{3}}{2} - \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)}\right)\right) + \left(\operatorname{re}{\left(x \left(- \frac{\sqrt[3]{3}}{2} + \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} + i \operatorname{im}{\left(x \left(- \frac{\sqrt[3]{3}}{2} + \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)}\right) 
=
    /                    2\       /                    2\                                 /                    2\     /                    2\
    |  /  3 ___      5/6\ |       |  /  3 ___      5/6\ |                                 |  /  3 ___      5/6\ |     |  /  3 ___      5/6\ |
    |  |  \/ 3    I*3   | |       |  |  \/ 3    I*3   | |    2/3            2/3           |  |  \/ 3    I*3   | |     |  |  \/ 3    I*3   | |
I*im|x*|- ----- + ------| | + I*im|x*|- ----- - ------| | + 3   *re(x) + I*3   *im(x) + re|x*|- ----- + ------| | + re|x*|- ----- - ------| |
    \  \    2       2   / /       \  \    2       2   / /                                 \  \    2       2   / /     \  \    2       2   / /
323re⁡(x)+re⁡(x(−332−356i2)2)+re⁡(x(−332+356i2)2)+323iim⁡(x)+iim⁡(x(−332−356i2)2)+iim⁡(x(−332+356i2)2)3^{\frac{2}{3}} \operatorname{re}{\left(x\right)} + \operatorname{re}{\left(x \left(- \frac{\sqrt[3]{3}}{2} - \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} + \operatorname{re}{\left(x \left(- \frac{\sqrt[3]{3}}{2} + \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} + 3^{\frac{2}{3}} i \operatorname{im}{\left(x\right)} + i \operatorname{im}{\left(x \left(- \frac{\sqrt[3]{3}}{2} - \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} + i \operatorname{im}{\left(x \left(- \frac{\sqrt[3]{3}}{2} + \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} 
producto
                            /    /                    2\     /                    2\\ /    /                    2\     /                    2\\
                            |    |  /  3 ___      5/6\ |     |  /  3 ___      5/6\ || |    |  /  3 ___      5/6\ |     |  /  3 ___      5/6\ ||
/ 2/3            2/3      \ |    |  |  \/ 3    I*3   | |     |  |  \/ 3    I*3   | || |    |  |  \/ 3    I*3   | |     |  |  \/ 3    I*3   | ||
\3   *re(x) + I*3   *im(x)/*|I*im|x*|- ----- - ------| | + re|x*|- ----- - ------| ||*|I*im|x*|- ----- + ------| | + re|x*|- ----- + ------| ||
                            \    \  \    2       2   / /     \  \    2       2   / // \    \  \    2       2   / /     \  \    2       2   / //
(323re⁡(x)+323iim⁡(x))(re⁡(x(−332−356i2)2)+iim⁡(x(−332−356i2)2))(re⁡(x(−332+356i2)2)+iim⁡(x(−332+356i2)2))\left(3^{\frac{2}{3}} \operatorname{re}{\left(x\right)} + 3^{\frac{2}{3}} i \operatorname{im}{\left(x\right)}\right) \left(\operatorname{re}{\left(x \left(- \frac{\sqrt[3]{3}}{2} - \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} + i \operatorname{im}{\left(x \left(- \frac{\sqrt[3]{3}}{2} - \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)}\right) \left(\operatorname{re}{\left(x \left(- \frac{\sqrt[3]{3}}{2} + \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)} + i \operatorname{im}{\left(x \left(- \frac{\sqrt[3]{3}}{2} + \frac{3^{\frac{5}{6}} i}{2}\right)^{2}\right)}\right) 
=
    3           2                  3             2         
9*re (x) - 27*im (x)*re(x) - 9*I*im (x) + 27*I*re (x)*im(x)
9(re⁡(x))3+27i(re⁡(x))2im⁡(x)−27re⁡(x)(im⁡(x))2−9i(im⁡(x))39 \left(\operatorname{re}{\left(x\right)}\right)^{3} + 27 i \left(\operatorname{re}{\left(x\right)}\right)^{2} \operatorname{im}{\left(x\right)} - 27 \operatorname{re}{\left(x\right)} \left(\operatorname{im}{\left(x\right)}\right)^{2} - 9 i \left(\operatorname{im}{\left(x\right)}\right)^{3} 
9*re(x)^3 - 27*im(x)^2*re(x) - 9*i*im(x)^3 + 27*i*re(x)^2*im(x)
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