Bessel
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25. y"+k2xy=0, dengan y=ux dan z=23kx32 Jawab: d2ydx2+k2xy=0 Misal y=ux12 dan z=23kx32 maka: dydx=u12x-12 +x12 dudx = u12x-12 +x12 dudzdzdz = u12x-12 +x12 kx12dudz =u12x-12 +kxdudz
d2ydx2=ddx12x-12 u+kxdudz = -14x-32 u+12x-12 dudx+kdudz+kxd2udz dx =-14x-32 u+12x-12 dudzdzdz+kdudz+kxd2udz2 dzdz =-14x-32 u+12x-12 kx12dudz+kdudz+kx kx12d2udz2 = -14x-32 u+12kdudz+kdudz+k2 x32d2udz2 =-14x-32 u+32kdudz+k2 x32d2udz2
Sehingga diperoleh: -14x-32 u+32kdudz+k2 x32d2udz2+k2xux12=0 -14x-32 u+32kdudz+k2 x32d2udz2+k2x32u=0 k2 x32d2udz2+32kdudz+k2x32u-14x-32 u=0 k2 x32d2udz2+32kdudz+k2x32-14x-32 u=0 __________________________________________________dikali49x32 49k2 x3d2udz2+23kx32dudz+49k2x3-19u=0 z2d2udz2+zdudz+z2-19u=0………Persamaan Bessel dengan variabel z dan v=13
Sehingga Solusi umum: ux=A0J13zx+B0Y13zx =A0J1323kx32+B0Y1323kx32
Karena y=ux, Jadi:yx=xA0J1323kx32+B0Y1323kx32
d2ydx2=ddx12x-12 u+kxdudz = -14x-32 u+12x-12 dudx+kdudz+kxd2udz dx =-14x-32 u+12x-12 dudzdzdz+kdudz+kxd2udz2 dzdz =-14x-32 u+12x-12 kx12dudz+kdudz+kx kx12d2udz2 = -14x-32 u+12kdudz+kdudz+k2 x32d2udz2 =-14x-32 u+32kdudz+k2 x32d2udz2
Sehingga diperoleh: -14x-32 u+32kdudz+k2 x32d2udz2+k2xux12=0 -14x-32 u+32kdudz+k2 x32d2udz2+k2x32u=0 k2 x32d2udz2+32kdudz+k2x32u-14x-32 u=0 k2 x32d2udz2+32kdudz+k2x32-14x-32 u=0 __________________________________________________dikali49x32 49k2 x3d2udz2+23kx32dudz+49k2x3-19u=0 z2d2udz2+zdudz+z2-19u=0………Persamaan Bessel dengan variabel z dan v=13
Sehingga Solusi umum: ux=A0J13zx+B0Y13zx =A0J1323kx32+B0Y1323kx32
Karena y=ux, Jadi:yx=xA0J1323kx32+B0Y1323kx32
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