Deret Pangkat 2
selesaikan persamaan dibawa ini dengan metode deret pangkat.
![\frac{d^2y}{dx^2}-2xy=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vd66lbuAEMvLWSuekFMXrZJO1wIyZ4WR-a5R0aJRU0fSzLJkgfxl6GE5tR3WoCOoLwuTpTaRv-DFzBLlTdcM47BPcGN43NFeHfigQeJUfXxRR1uHyaTexsrJegxUZb9O0MV2yI_2-Yd-vEFLE=s0-d)
penyelesainnya mirip dengan yang sebelumnnya, yakni kita misalkan.
![y=\sum_{n=0}^{\infty}a_nx^n](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sd43XJ4Lz95pl5TIEi3hHc86_7lSsJsHmCDCELG9tD3TX1XkGi5LiagnEsv436Ljs5WARkEnjJpFho_aoadN_eAgxT9fJpLSmM4hL5Ud2cGBeC4bOWw8l2j7l3lEFpqodhpCZG71brLNNwnup7W5zr-hLUCg=s0-d)
![y'=\sum_{n=1}^{\infty}na_nx^{n-1}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uXHW8E6stua_RtprlGFj7jlxvpGOH15AUBIRq_5QDINSwPYTLuVeQvPBirQmBaaU7Orxxv4C3dKYn4fPxSWvAHUgsgQl192aP5YBRzzoVKr4B1p7RwpxBZCau4FeIJLfEAXVcSO_keSwq3DS13NH0PUUZnBvcH7N5RfNKOwxg=s0-d)
![y'=\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tmGZ5JrMqJQ_MAOPqoKYUoyakHVGsga-xuipHdOJuzzAp810CnE2XmcpEYV04BI_NOHu08xSjPJ8OoxPlQUgAj8erDQ_LltzcS83fW5N4yDE4t6KNXGVe2GsZnvjIc3Q8Fn71akhrf1i5SakhXYVNa7-86izE9XBp3-HkVIBXMbFL3=s0-d)
![\frac{d^2y}{dx^2}-2xy=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vd66lbuAEMvLWSuekFMXrZJO1wIyZ4WR-a5R0aJRU0fSzLJkgfxl6GE5tR3WoCOoLwuTpTaRv-DFzBLlTdcM47BPcGN43NFeHfigQeJUfXxRR1uHyaTexsrJegxUZb9O0MV2yI_2-Yd-vEFLE=s0-d)
![\sum_{n=2}^{\infty}n(n-1)a_nx^{n-2}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_s8xHfRCyPOyWrUqv-SCdAwhDDsLmZ1LZ07zOFydixL7fR5fhUAAihHORKd7PHGxYqr-ENFjXaWrLaD_4DlQO7SMQrLdw5qRygMbQJY0dHfWRGua7cXfK6gFQ991fQxB-Okv5aaQsduX9XMQevWOQxm9WmkYXEYyzfY_2bI5w=s0-d)
![+2x\sum_{n=0}^{\infty}a_nx^n=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u6DZUEAsW4pAwUP5GSsTJ8FsX7aTn9Qetx368zqoeJjTppl3O0_tFoYWYwnpER5iSSObw7hMP92x71rsMRZg2ip9mpnrSrb9WC4TSIBzWfg9JIo8dbiY93AmCdSNoIN5SwIZBM-s19A39_yz8cLePSBdWvwxk=s0-d)
![\sum_{n=2}^{\infty}n(n-1)x^{n-2}+2\sum_{n=0}^{\infty}a_nx^{n+1}=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t1W-RwU2F8lqjxmY7cw3kjCn5EAbYdISAK2HR_AcS4sE_V1eWius0SgNEcO0317yRgemaV3a1zn971rqjvdnNPKZdt3ceFP7FxertliGsHp-Y2i3UUpnLkipQERapBhJHGeiTy0mELLQogFF_PQVAFo2_bknp5shFEGOPRw2geDooW1eWXxKZMqYiluJRN8Ivv1YkiXi6BBh8K848aavSQWUkk5xjXUKxEkjCWU3f1oE8fq6A=s0-d)
kita misalkan
dan
![k=n+1\rightarrow n=k-1](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_s1jXYK3kWsfsgrnMrAfb4Ce2LcfOUu3bynXBn5FlBvGIiRMH3ZpO4_JCTXc1_m-Is007-aNT92TCWcl_OnRrYp6r5IpnXfpbLzWlACts1UBgVVvP5I4wwg9I6FHZ2zVQS_uNlA5g=s0-d)
![\sum_{k=0}^{\infty}(k+2)(k+1)a_{k+2} x^k+2\sum_{k=1}^{\infty}a_{k-1}x^k=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v9crCauFrd5-UlgJCLQs1ED9___Ibepkad2N9H0Jtcd0jDQlzOoQ2pGtn3fmRnrxpOPpfCSS8otb6BpyLlJWduqP4WSvMlfAlQ1dVHtv3DlZIRYeoMB54ZJuiH8nUMk0z7kz8-peSUwDaTcKWpRn5tAz-IZddUJvomgtTdWHvd9Onfc-9YGgaZEy--fEuBadP2033BYk8_fv0VjBxCoBFcixtaXvVjNGGKAk_KtukmjRe9hNZcVrxaHBi_LPM4ZM4Z=s0-d)
jika
![k=0\rightarrow (2)(1)a_2 x^0=2a_2](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sSHVXJ4hfOPUhIbzzzg_pDeVHvzjn396JM_dmas7sURghfRPHtrAAKnJL1JyRQWAlFsdFoK7__HnwQap4dfyLFBkGnwD9aSSs6pSwYXhOGimy_iYKdPVZmqJZGvwZE9Ur4NJiYctJoQpItbxHRp0_2uRlcJxNIUQ=s0-d)
![2a_2+\sum_{k=1}^{\infty}(k+2)(k+1)a_{k+2}x^k+2\sum_{k=1}^{\infty}a_{k-1}x^k=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_twa83ZqiruVnttdwCvUJgbdBWnZzG6SCH-VTGhqRP8TxCZlS7u5NGqatbXu7RK52OlZIm2c9hkhdmkXpSwdbd19LlR1LouGCmIu2DDwykwKUJ2qBHmEmJkCDOE867aJ-P2mYaywGnye0zNYIvm6-G47kDtr4vS_UIicS8twrFqteMzaWWTVqxCKKq3NzGd-VRASRHw_xi3TbMUxyXXEiBbVfADNB0ii0erpHGpQEBcA3zJ4J-HV1OjR6mJccFsRmY=s0-d)
![2a_2+\sum_{k=1}^{\infty}[(k+2)(k+1)a_{k+2}x^k+2a_{k-1}]x^k=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vOrMEeIsUlXwNG7Ay5zgL3hXQbQXRwGPlgY6s9dlpM18iRxqkL-lJzmatOakga7LTyCyHqJncTmb2Y9yviRH4v0lF_ZXoa4ydxVYl7ChNzIlTwW2suB1fIeHNCXHuBjbeBXFo20GRFo8ysLjMxbNFB4FaszQndwbQTp9oIiVo-PtsXITvEjefOMb75iWVJyGLB5F1zbhsKAkBWnMfFF2MV0-iCig=s0-d)
jelas bahwa
![2a_2=0 \rightarrow a_2=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sdGB1D2q_h4dTllXYP52W3R2nzNbmmn21xKm60Y77bENN6edvui_UK-ZFlWzcI2wOaNDwrwZVKlNKN-Iw1zcRPM0tcA608uTd64eWit6cYIF7v1mhRLBTdlxojc6CV_mvN-wsso5Cl54uzRIs=s0-d)
juga komponen yang harus nol adalah
![(k+2)(k+1)a_{k+2}+2a_{k-1}=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uAfJd-4GGBFC4Qn3TDXvzDR8nLbOzCpKLwx8F366D6-expD7IVk0EG7KjWekIwhtwNhhezoPA0S-26Lxr3IWr0R9TsRHW5LPdmeTWgusnkEaU3bWj5vjd7IQ_gZUpTzhrnwXjnc7NIrI35=s0-d)
![a_{k+2}=-\frac{2a_{k-1}}{(k+2)(k+1}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sxs_7OK_krwDz8KbryP-kMut1cTYDXYkiGhvqHAt26nj4mtHdltigwDbnt8ikBL3NCszeZVWFjBCtWaXBIxPaAr5tVkLF4XitsVZjy5GDZozsxGgn5wBqn4LLHfErkN7mfb7toMXw7E8x8xOS0un75af2yxiGa4ckstoqi=s0-d)
sekarang kita subtitusi nilai
sampai
maka di peroleh
![a_2=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uDIJ3VtVd3jFmshxHBvpoLWYELA8ACc4AxiCbydoxBCV2eNe3qR_oTxevmJeTjw56KR-gZzYDL1UVdrp98xgp-YrRlbGiRkdAzUoor=s0-d)
![a_3=-\frac{a_0}{3}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t3KwHE3dVJsNo2ltIJbWS5_3f94UNDuZJoEywVPJhQJj2JBIujQwscfLaW0JlLXLaRe_RZxOj74h1bf55ZE9sU-tQCVRDMEYZId19x_64d-L5k2kt5VvOzoBHbZNzmm_pMItA=s0-d)
![a_4=-\frac{a_1}{6}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vTOf4XVHapzKhl0h6gjt3mys7_txE_tLqq7IgqUL9nF49hzKuPhPr9-ETQL1lWwkxDOn9m7nBQhERTjeeuhVAgO4bbn-cUOXNqn1fdQh82RF8bhRD2OpU7LnyMHYNObYKDhA=s0-d)
![a_5=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t1x7AL6VNsZsFX4mQmF8t28RrcZkMpcuaAUA0otPpWub9p2ue7I0Ycf3JE-1-TJXq_HE7hx_DjrR-JigPSl2_AhEB4aO58EGd2Ixug=s0-d)
![a_6=\frac{a_0}{45}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uyN04a6TDcmDOSgFwcsl33C5QzDIMWl-ph5Hg9JBmjasVkhDMpCRD-MUoNUgfs7VFOACN2opBsd9bOdSPn6H6rYI-ZX9beVdtoYl1bk0PqJUxEj0R8VCvat50am3xiSB7lNbc=s0-d)
![a_7=\frac{a_1}{126}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u47SbtUCC8OPNhRulJRxSaRzHg1L3DBSFGVF_7CsXO0NGZapshagit4cY5uG0OActwRQ32qJbQ2ZX-JoEdsR6B-35iLS2ypGRowclvgzXDv8_LMfZFtHe6m0mbKawJwOGFvZc0=s0-d)
![a_8=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u2PjNJuDwyFmZ4hYzI-JgfnhMLSHaCkJMZeDnkK7EUtDaP3f88Raif5BKMPvb6CAh-veJ_MMruCwTYS4u66_YSLQjgUaSGROiQKZGI=s0-d)
![a_9=\frac{a_0}{1620}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u0dUcmbWYLJJFJMt9dqd2ZyTr-7Qyeu9bElRs39FJldyMNtFqt3Xyd-vofMkX473VQ0fnZPL9V7uclVaMeZtLvnzz5FfCJFM3z4NI2zeeijhbbPjP6vIDVYTUoxumfbkTZMQhZ6w=s0-d)
dengan demikian diperoleh 2 solusi yaitu
1.![a_0=1\rightarrow a_1=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v5g-fg8-vkGd1mitKgBlQw9RDs7FPhQvlvbkPaqbBcMpioHe6ncyhvduNHD-rc90Akq4Wqequ6jiEnLrfS7odpwXVsg9re6rt4EzAqz8JDT-HVVVJ72rBdahkyNcHXH9UR6wsPJA=s0-d)
2.![a_0=0\rightarrow a_1=1](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_syOwnFN9fVUMaD2IQD5-2g28HEhpL8pDxwYd7rM38QrFJ3FUkwc8veJYoXNxkVGfVQ75RooIL07ur91NnbEmNvFL1-d3saQTI1WEbSHkSP6hq9Cv2YaTLx2NlC9PBh5oR8dm0-=s0-d)
untuk kasus 1
![y_1(x)=1-\frac{1}{3}x^3+\frac{1}{6}x^6-\frac{1}{1620}x^9+..........](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uOuzUWhIpm9oMNNx8ep6y0q1-y-N5VV4mmk0dyu_iHr_fGrlZzobZKbrtllg4Y_vFKstvcdohDJ4OfN6xudjZ-FgpOoYg2cSQpObFsOGcT9xDPNtZaQsDmsk9kXmowmKmSCWR-Cwk4Foi9nlR2G_2_pPKYtiVfbgF9tdmFXdJr8ou8HMw8YQgYT3VkvkMRsaSSLYa3QG7wGdsIcLmzEqdMT5E23qtG3tomUJ9OPFk=s0-d)
untuk kasus 2
![y_2(x)=x-\frac{1}{6}x^4+\frac{1}{126}x^7+..........](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vAW2ufRvymcUEenVnaz4fPw0rrshTr4h85xLInfZj-gGXbGXJkEq6H3nm9cIsFUuIj4F8FkYxThE7UVuJUvh61MePn7R9lSQBQJH2Ic14H71e2IOrOSbg-Vw8P4oPDr-Rl0XGaoIk_JxEKSQDWvNbHulD0Aufp_stdI5i8vHj6FMe69erLuBCu2ukklP-A5XVwRQ=s0-d)
dengan demikian dapat ditulis secara lengkap
![y_1(x)=a_0[1-\frac{1}{3}x^3+\frac{1}{6}x^6-\frac{1}{1620}x^9..........]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_veunCaMgojGb3FK1GVouH9OFa7CLgE6vhDV31Pe-SQWIJ-fWj3Fyg_sWieZUegauYELVNW6JSvbn8T02gM0vCVuclipSQMAXm0Ia042kR8SeZePIxuAq8lBl58jPbkKk-aoCyDNrHLB6jXU3BHot5UbJqAOI6ES3e6-_voX2Xx9faiA5mlPIlJi_MXSk3rW2hTqMSbE_Lr2XEHnW95dGWVVyIZFoIBZrQE2A7eEuigND5-UEYiVFc=s0-d)
![y_2(x)=a_1[x-\frac{1}{6}x^4+\frac{1}{126}x^7+..........]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_u1xPP1CVdSEHsxH59_dZIgrRQ_9iMfn4_0fzQ-GjmSrLMkoGqGQZ42DyFhD9aWhyJ_-qHiGEc2__TGCUg56sVB-iDJqx_FJ__tk3RN9RdDAhOo1cjWLqWCInt8zwbMaT5VtQ7yYviEkjNKdW6nHxbeCuRyU06dcQkfQl_1Gdpg1aRO4Q-2J_adpuaBtJhrQnOjmTRKs7QaVlm6sw=s0-d)
saya kira ini mudaji...
dan kalau ada pertanyaan, silahkan coment, pertannyaannya tentang tugas kulia juga boleh.
hehehehe nanti kita kerjakan sama-sama.
penyelesainnya mirip dengan yang sebelumnnya, yakni kita misalkan.
kita misalkan
jika
jelas bahwa
juga komponen yang harus nol adalah
sekarang kita subtitusi nilai
dengan demikian diperoleh 2 solusi yaitu
1.
2.
untuk kasus 1
untuk kasus 2
dengan demikian dapat ditulis secara lengkap
saya kira ini mudaji...
dan kalau ada pertanyaan, silahkan coment, pertannyaannya tentang tugas kulia juga boleh.
hehehehe nanti kita kerjakan sama-sama.
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