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sin(((sqrt2mE)l)/h)=sqrt(E)/(U) la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   /  _____    \     ___
   |\/ 2*m *E*l|   \/ E 
sin|-----------| = -----
   \     h     /     u
sin⁡(le2mh)=eu\sin{\left(\frac{l e \sqrt{2 m}}{h} \right)} = \frac{\sqrt{e}}{u}
Simplificar
Solución detallada
Tenemos la ecuación:
sin⁡(le2mh)=eu\sin{\left(\frac{l e \sqrt{2 m}}{h} \right)} = \frac{\sqrt{e}}{u}
Usamos la regla de proporciones:
De a1/b1 = a2/b2 se deduce a1*b2 = a2*b1,
En nuestro caso
a1 = 1

b1 = 1/sin(E*l*sqrt(2)*sqrt(m)/h)

a2 = 1

b2 = u*exp(-1/2)

signo obtendremos la ecuación
ue12=1sin⁡(2elmh)\frac{u}{e^{\frac{1}{2}}} = \frac{1}{\sin{\left(\frac{\sqrt{2} e l \sqrt{m}}{h} \right)}}
ue12=1sin⁡(2elmh)\frac{u}{e^{\frac{1}{2}}} = \frac{1}{\sin{\left(\frac{\sqrt{2} e l \sqrt{m}}{h} \right)}}
Abrimos los paréntesis en el miembro izquierdo de la ecuación
u*exp-1/2 = 1/sin(E*l*sqrt(2)*sqrt(m)/h)

Abrimos los paréntesis en el miembro derecho de la ecuación
u*exp-1/2 = 1/sin*l*sqrt+2sqrtmh)

Dividamos ambos miembros de la ecuación en exp(-1/2)
u = 1/sin(E*l*sqrt(2)*sqrt(m)/h) / (exp(-1/2))

Obtenemos la respuesta: u = exp(1/2)/sin(E*l*sqrt(2)*sqrt(m)/h)
Gráfica
Suma y producto de raíces [src] 
suma
                              /          /    ___\\         /          /    ___\\                                                           /          /    ___\\          /          /    ___\\                         
                              |    ___   |l*\/ m ||  1/2    |    ___   |l*\/ m ||                                                           |    ___   |l*\/ m ||  1/2     |    ___   |l*\/ m ||                         
                          cosh|E*\/ 2 *im|-------||*e   *sin|E*\/ 2 *re|-------||                                                      I*cos|E*\/ 2 *re|-------||*e   *sinh|E*\/ 2 *im|-------||                         
                              \          \   h   //         \          \   h   //                                                           \          \   h   //          \          \   h   //                         
----------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------
    /          /    ___\\      /          /    ___\\        /          /    ___\\     /          /    ___\\       /          /    ___\\      /          /    ___\\        /          /    ___\\     /          /    ___\\
   2|    ___   |l*\/ m ||     2|    ___   |l*\/ m ||       2|    ___   |l*\/ m ||    2|    ___   |l*\/ m ||      2|    ___   |l*\/ m ||     2|    ___   |l*\/ m ||       2|    ___   |l*\/ m ||    2|    ___   |l*\/ m ||
cos |E*\/ 2 *re|-------||*sinh |E*\/ 2 *im|-------|| + cosh |E*\/ 2 *im|-------||*sin |E*\/ 2 *re|-------||   cos |E*\/ 2 *re|-------||*sinh |E*\/ 2 *im|-------|| + cosh |E*\/ 2 *im|-------||*sin |E*\/ 2 *re|-------||
    \          \   h   //      \          \   h   //        \          \   h   //     \          \   h   //       \          \   h   //      \          \   h   //        \          \   h   //     \          \   h   //
e12sin⁡(2ere⁡(lmh))cosh⁡(2eim⁡(lmh))sin⁡2(2ere⁡(lmh))cosh⁡2(2eim⁡(lmh))+cos⁡2(2ere⁡(lmh))sinh⁡2(2eim⁡(lmh))−ie12cos⁡(2ere⁡(lmh))sinh⁡(2eim⁡(lmh))sin⁡2(2ere⁡(lmh))cosh⁡2(2eim⁡(lmh))+cos⁡2(2ere⁡(lmh))sinh⁡2(2eim⁡(lmh))\frac{e^{\frac{1}{2}} \sin{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}}{\sin^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} + \cos^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}} - \frac{i e^{\frac{1}{2}} \cos{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}}{\sin^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} + \cos^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}} 
=
                              /          /    ___\\         /          /    ___\\                                                           /          /    ___\\          /          /    ___\\                         
                              |    ___   |l*\/ m ||  1/2    |    ___   |l*\/ m ||                                                           |    ___   |l*\/ m ||  1/2     |    ___   |l*\/ m ||                         
                          cosh|E*\/ 2 *im|-------||*e   *sin|E*\/ 2 *re|-------||                                                      I*cos|E*\/ 2 *re|-------||*e   *sinh|E*\/ 2 *im|-------||                         
                              \          \   h   //         \          \   h   //                                                           \          \   h   //          \          \   h   //                         
----------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------
    /          /    ___\\      /          /    ___\\        /          /    ___\\     /          /    ___\\       /          /    ___\\      /          /    ___\\        /          /    ___\\     /          /    ___\\
   2|    ___   |l*\/ m ||     2|    ___   |l*\/ m ||       2|    ___   |l*\/ m ||    2|    ___   |l*\/ m ||      2|    ___   |l*\/ m ||     2|    ___   |l*\/ m ||       2|    ___   |l*\/ m ||    2|    ___   |l*\/ m ||
cos |E*\/ 2 *re|-------||*sinh |E*\/ 2 *im|-------|| + cosh |E*\/ 2 *im|-------||*sin |E*\/ 2 *re|-------||   cos |E*\/ 2 *re|-------||*sinh |E*\/ 2 *im|-------|| + cosh |E*\/ 2 *im|-------||*sin |E*\/ 2 *re|-------||
    \          \   h   //      \          \   h   //        \          \   h   //     \          \   h   //       \          \   h   //      \          \   h   //        \          \   h   //     \          \   h   //
e12sin⁡(2ere⁡(lmh))cosh⁡(2eim⁡(lmh))sin⁡2(2ere⁡(lmh))cosh⁡2(2eim⁡(lmh))+cos⁡2(2ere⁡(lmh))sinh⁡2(2eim⁡(lmh))−ie12cos⁡(2ere⁡(lmh))sinh⁡(2eim⁡(lmh))sin⁡2(2ere⁡(lmh))cosh⁡2(2eim⁡(lmh))+cos⁡2(2ere⁡(lmh))sinh⁡2(2eim⁡(lmh))\frac{e^{\frac{1}{2}} \sin{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}}{\sin^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} + \cos^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}} - \frac{i e^{\frac{1}{2}} \cos{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}}{\sin^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} + \cos^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}} 
producto
                              /          /    ___\\         /          /    ___\\                                                           /          /    ___\\          /          /    ___\\                         
                              |    ___   |l*\/ m ||  1/2    |    ___   |l*\/ m ||                                                           |    ___   |l*\/ m ||  1/2     |    ___   |l*\/ m ||                         
                          cosh|E*\/ 2 *im|-------||*e   *sin|E*\/ 2 *re|-------||                                                      I*cos|E*\/ 2 *re|-------||*e   *sinh|E*\/ 2 *im|-------||                         
                              \          \   h   //         \          \   h   //                                                           \          \   h   //          \          \   h   //                         
----------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------
    /          /    ___\\      /          /    ___\\        /          /    ___\\     /          /    ___\\       /          /    ___\\      /          /    ___\\        /          /    ___\\     /          /    ___\\
   2|    ___   |l*\/ m ||     2|    ___   |l*\/ m ||       2|    ___   |l*\/ m ||    2|    ___   |l*\/ m ||      2|    ___   |l*\/ m ||     2|    ___   |l*\/ m ||       2|    ___   |l*\/ m ||    2|    ___   |l*\/ m ||
cos |E*\/ 2 *re|-------||*sinh |E*\/ 2 *im|-------|| + cosh |E*\/ 2 *im|-------||*sin |E*\/ 2 *re|-------||   cos |E*\/ 2 *re|-------||*sinh |E*\/ 2 *im|-------|| + cosh |E*\/ 2 *im|-------||*sin |E*\/ 2 *re|-------||
    \          \   h   //      \          \   h   //        \          \   h   //     \          \   h   //       \          \   h   //      \          \   h   //        \          \   h   //     \          \   h   //
e12sin⁡(2ere⁡(lmh))cosh⁡(2eim⁡(lmh))sin⁡2(2ere⁡(lmh))cosh⁡2(2eim⁡(lmh))+cos⁡2(2ere⁡(lmh))sinh⁡2(2eim⁡(lmh))−ie12cos⁡(2ere⁡(lmh))sinh⁡(2eim⁡(lmh))sin⁡2(2ere⁡(lmh))cosh⁡2(2eim⁡(lmh))+cos⁡2(2ere⁡(lmh))sinh⁡2(2eim⁡(lmh))\frac{e^{\frac{1}{2}} \sin{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}}{\sin^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} + \cos^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}} - \frac{i e^{\frac{1}{2}} \cos{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}}{\sin^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} + \cos^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}} 
=
                                                    1/2                                                  
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    /          /    ___\\    /          /    ___\\        /          /    ___\\     /          /    ___\\
    |    ___   |l*\/ m ||    |    ___   |l*\/ m ||        |    ___   |l*\/ m ||     |    ___   |l*\/ m ||
cosh|E*\/ 2 *im|-------||*sin|E*\/ 2 *re|-------|| + I*cos|E*\/ 2 *re|-------||*sinh|E*\/ 2 *im|-------||
    \          \   h   //    \          \   h   //        \          \   h   //     \          \   h   //
e12sin⁡(2ere⁡(lmh))cosh⁡(2eim⁡(lmh))+icos⁡(2ere⁡(lmh))sinh⁡(2eim⁡(lmh))\frac{e^{\frac{1}{2}}}{\sin{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} + i \cos{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}} 
exp(1/2)/(cosh(E*sqrt(2)*im(l*sqrt(m)/h))*sin(E*sqrt(2)*re(l*sqrt(m)/h)) + i*cos(E*sqrt(2)*re(l*sqrt(m)/h))*sinh(E*sqrt(2)*im(l*sqrt(m)/h)))
Respuesta rápida [src] 
                                   /          /    ___\\         /          /    ___\\                                                           /          /    ___\\          /          /    ___\\                         
                                   |    ___   |l*\/ m ||  1/2    |    ___   |l*\/ m ||                                                           |    ___   |l*\/ m ||  1/2     |    ___   |l*\/ m ||                         
                               cosh|E*\/ 2 *im|-------||*e   *sin|E*\/ 2 *re|-------||                                                      I*cos|E*\/ 2 *re|-------||*e   *sinh|E*\/ 2 *im|-------||                         
                                   \          \   h   //         \          \   h   //                                                           \          \   h   //          \          \   h   //                         
u1 = ----------------------------------------------------------------------------------------------------------- - -----------------------------------------------------------------------------------------------------------
         /          /    ___\\      /          /    ___\\        /          /    ___\\     /          /    ___\\       /          /    ___\\      /          /    ___\\        /          /    ___\\     /          /    ___\\
        2|    ___   |l*\/ m ||     2|    ___   |l*\/ m ||       2|    ___   |l*\/ m ||    2|    ___   |l*\/ m ||      2|    ___   |l*\/ m ||     2|    ___   |l*\/ m ||       2|    ___   |l*\/ m ||    2|    ___   |l*\/ m ||
     cos |E*\/ 2 *re|-------||*sinh |E*\/ 2 *im|-------|| + cosh |E*\/ 2 *im|-------||*sin |E*\/ 2 *re|-------||   cos |E*\/ 2 *re|-------||*sinh |E*\/ 2 *im|-------|| + cosh |E*\/ 2 *im|-------||*sin |E*\/ 2 *re|-------||
         \          \   h   //      \          \   h   //        \          \   h   //     \          \   h   //       \          \   h   //      \          \   h   //        \          \   h   //     \          \   h   //
u1=e12sin⁡(2ere⁡(lmh))cosh⁡(2eim⁡(lmh))sin⁡2(2ere⁡(lmh))cosh⁡2(2eim⁡(lmh))+cos⁡2(2ere⁡(lmh))sinh⁡2(2eim⁡(lmh))−ie12cos⁡(2ere⁡(lmh))sinh⁡(2eim⁡(lmh))sin⁡2(2ere⁡(lmh))cosh⁡2(2eim⁡(lmh))+cos⁡2(2ere⁡(lmh))sinh⁡2(2eim⁡(lmh))u_{1} = \frac{e^{\frac{1}{2}} \sin{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}}{\sin^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} + \cos^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}} - \frac{i e^{\frac{1}{2}} \cos{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}}{\sin^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \cosh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} + \cos^{2}{\left(\sqrt{2} e \operatorname{re}{\left(\frac{l \sqrt{m}}{h}\right)} \right)} \sinh^{2}{\left(\sqrt{2} e \operatorname{im}{\left(\frac{l \sqrt{m}}{h}\right)} \right)}} 
u1 = exp(1/2)*sin(sqrt(2)*E*re(l*sqrt(m)/h))*cosh(sqrt(2)*E*im(l*sqrt(m)/h))/(sin(sqrt(2)*E*re(l*sqrt(m)/h))^2*cosh(sqrt(2)*E*im(l*sqrt(m)/h))^2 + cos(sqrt(2)*E*re(l*sqrt(m)/h))^2*sinh(sqrt(2)*E*im(l*sqrt(m)/h))^2) - i*exp(1/2)*cos(sqrt(2)*E*re(l*sqrt(m)/h))*sinh(sqrt(2)*E*im(l*sqrt(m)/h))/(sin(sqrt(2)*E*re(l*sqrt(m)/h))^2*cosh(sqrt(2)*E*im(l*sqrt(m)/h))^2 + cos(sqrt(2)*E*re(l*sqrt(m)/h))^2*sinh(sqrt(2)*E*im(l*sqrt(m)/h))^2)
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