Deret Pangkat
Selesaikanlah persamaan dibawa ini dengan metode deret pangkat
![y'+ay=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sTrrpfKu_2qEzTLJsHbU1yDLyHj1i3y2PNAl62Uyn-NwTXixKyn5Xkm95XOjaJh1D749lMX0GL9_QwDq1zr34gQyMiFirO9L0RWsppoSc=s0-d)
Untuk menyelesaikan soal diatas, kita misalkan
![y=\sum_{n=0}^{\infty }a_nx^n](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_umL9TSB8WnwQpiuBWHg27g86tQW90J-pclNiw59yKbbd6KiohNjLZtO1c5JaElqtqkDZIJTigVDCK4o7HIArwaIpf2Y0mKDNfVPufR-jw2NAv3eDJAm9dfkfeU5ae_SrKk1jINvpIm33fuuAZetGbxCu8ryE33FyO__eo=s0-d)
![y'=\sum_{n=0}^{\infty }na_nx^{n-1}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ukVgW9vyzvqASpda7AJC55A3qx2yQHakKQ9dYCmriyOcVw2owBiFxvtkzqy91fLKd15RP_P9_uwylQvrCnCTNHzhgqN8_qYkb8aKivltm5WgRyJSeTtEfCOXEGAx74wK4ZEuZi7o-9T1YSdjGsfTr0TdWmY95KZEwK1-JFrDaGG9iDFSi0=s0-d)
![y'+ay=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sTrrpfKu_2qEzTLJsHbU1yDLyHj1i3y2PNAl62Uyn-NwTXixKyn5Xkm95XOjaJh1D749lMX0GL9_QwDq1zr34gQyMiFirO9L0RWsppoSc=s0-d)
![\sum_{n=1}^{\infty }na_nx^{n-1}=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_udwGPU1oxfURl_sUP0Q-ofRex5S0Q5RRxviHeL5nVXuvcTpj6W92JpixaOx7DlLfER01d2PBg-uGpaDXnel_c308G3_jgEJafZZxyO1NlYqTikaTLqO--nE-EqOlR6XJzZfhlfPICUVolKAT8D2KBmlztAaSX1VCGyf8Sdn203FsNT=s0-d)
![+a\sum_{n=0}^{\infty }a_nx^n=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uvfmZVKe_JWCM1g-tTo9OVWxZmHO54WLQlXktTqd_E1aF1uKrTHA9h1_VpPNTWc2MVNv9J_Vit_12ZPK1Aj8Q-qmS1x6OiH-ID4W3LTRGpHtlLcWDHgwFUD7a5tH33xCCtPOg0RwCskY9lu0kgucE-6pk6C3yzP3mC0vQ=s0-d)
miaslkan![n=n+1](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_t9cCEFK-erkhUYSZhVLRUlhqxaTNkXfSSi_5DDhJivjiqBrcNJ4DojpbyCE6skbUucn6769IzrvRz5i95VxWDxQeEhiP9xP9KHsbaJ=s0-d)
![\sum_{n=0}^{\infty }(n+1)a_{n+1}x^{n+1-1}+a\sum_{n=0}^{\infty}a_n x^n=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tuw6l1qRCwxCvhWQLGIDCwCK2ekkfTDlo5AFJyMmGhSZsRJHcWiZ9_3PYgJfz46-ucMT31P3fjiftccSvBcvmhhWGbVQEw2NzmYifMAE_DuoA-0Ierd8IUjkyrGuIMKjqchlzKJ43iL5dqI3Hm6YOziuBPhRZ50r1O53YooRAbIGUpMYGBevogkBgjkck1N5ALJOmxU4mtyUwUAEbAp49lkftXBbr-M6p7gQJeUr8-me4PSTZHzllooX5s2LmiNF3mYA=s0-d)
![\sum_{n=0}^{\infty }[(n+1)a_{n+1}x^n+aa_n x^n]=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tf0Jz_Xk03bu4sbx8_qwlxGaFAc0HIQNka8G_tbuOQtIEYzh0HNJFCapCZMoesJKZMW3UzbBthL--MGiGMtqrSYPNhV1_SmRnBuFJNydPOIhaOMx-bbFt3_FPKFaxHMo6-uTd8LZVgO_vEttNAStyatyBjhj6pHsgO3jgt66s_Uu6uU_TxM4VRCadpfRkGMImCT4tYOmv8ER_ljYA=s0-d)
![\sum_{n=0}^{\infty }[(n+1)a_{n+1}+aa_n]x^n=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uS0gEnddlSLgFjdCsV3njCHuu4ThS3B3F4exODVD8lWkjzlSCHXJ4_X_7DkjQ9LNYKNh6mgMxFKTL8RZ8WcZJ41qqJrOcab5X25b2z_ZaYlgThQlCQP1eUjJ0uQzMDw36z_pI8GZMd_QXWv0HQgXs6ff5fRXhHGcCkjjRQoNyu7CY5h7uWbqIQZj84nZs6iZM=s0-d)
karna
bernilai sembarang dan tidak mungkin sama dengan 0 maka yang harus 0 adalah suku yang ada di dalam kurung.
![(n+1)a_{n+1}+aa_n=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ur8-GP9o26LCOxoAEkeWTVWoIvneZqTmpQ5JrbhRNp_vu-z_HlkCzesH8NULOODOv1eJumKJkxMZ3gaKsEmKtVWLZ2057LjkIQXlr3wzmlgXYGWJtnXUj2v8Q6AwH7=s0-d)
![a_{n+1}=-\frac{aa_n}{n+1}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tO-chysx6Tl5l7evI6h1k4lBMmRqgnwAcxHqVnJZn8hvvmI_H1ignn_twbnDX6l0w7HaOF4PxmABsT0ff_dP1ooAVbJrdFDiGEUqwg1Ob6C-mTL6WYyJfFTxNgVPJO005dDNhyp3FGn1WleWTK=s0-d)
sekarang kita hitung untuk
sampai ![n=3](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vjiU6YBlzmCEtGJj0oHmd1bKt0MV2Y5_tHn_biqM-NtDQ3NoGbhaO4BpUn2IZ_9qe1CIcRfDUpIIZDmvFv4z-mefi1amxOuzR-Sw=s0-d)
![a_{n+1}=-\frac{aa_n}{n+1}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tO-chysx6Tl5l7evI6h1k4lBMmRqgnwAcxHqVnJZn8hvvmI_H1ignn_twbnDX6l0w7HaOF4PxmABsT0ff_dP1ooAVbJrdFDiGEUqwg1Ob6C-mTL6WYyJfFTxNgVPJO005dDNhyp3FGn1WleWTK=s0-d)
![n=0\Rightarrow a_1=-\frac{aa_0}{1}\Rightarrow a_1=-aa_0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tlv0ys86qVL7GTpUyshvpvsz11Ar-dzvMrlvzPlfDzO4gL-Ow6IuhNxirDCaxy4pSx0RQMErSCYPXR2UvOGYsAnrOdM3Uv5jYsda1ewW2lTAVnAHL37iKBsdwtX-pDA0L3k0yM1Na7_b7tfObhJd0Etv4pyf7TICFlgL_P3Xg8bSW6gUXPq7GIIwok85aWEl8PD6qxWWGZsQ=s0-d)
![n=1\Rightarrow a_2=-\frac{aa_1}{2}\Rightarrow a_1=\frac{a^2a_0}{2!}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tGqbAna0Bt-bvin5FYOjUOQ0E9onLnu_hS83QHInx5SAfcEObMQHse3BMlc0hoZGHor8eCoe2KE5sPF7I9x_jIn51isejlqu2iZtV42TkkNq9qX5_AN8yz0JI3t4FLzioj9K7nWyJlf8NclZcmWMuOG_cdTe8ZyIIbuoH8hmIb8u-oEjGX6lhbHzgXwPLVX3kK6ztnNpdvGHOuOS-DEoy996W_h7FqEhI3RRzExG-Yi-g=s0-d)
![n=2\Rightarrow a_3=-\frac{aa_2}{3}\Rightarrow a_3=-\frac{a^3a_0}{3!}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v70qe7mhYNtJKDhlNwmZydrDghnCs16YZ0Jb23HbEIKHbT9QHgC8wwJVVERrhCIDazWQLdfEXJEIybcBAOiQWVtMtfB5jdzvItjtt6_66obIQNz5gSUs4vVa7V7ftnBYDzTP6vbfkFr1FjGxQ3sSQjLAEispkqJfez2WN3XSqNB2v3EO-cRba8h6lKAYCkatX4uhs0nG7zjuU3if0ZSj0akTTgntTkLxKopq3X5MS0nJo=s0-d)
![n=3\Rightarrow a_4=-\frac{aa_3}{4}\Rightarrow a_4=-\frac{a^4a_0}{4!}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tQLTzau03DLzQKAtORLJdgVdibg6z4CTLpYJqa7fU0DWmo6RJCWSrkGdKbsaW8Y-nFnMJhGFH7VDISInR7Z6R8298D8AXLENbflcfZCi-PVRavguen82_BUeeNEBwkAlIk81sUjykrqLgG1mEkrfvc26pr98ty0oNZqL2qwP3DJRjvDA_38zNX5iI0947pDbUcnc1L9ZfQGAIeMoQ0UsNbyZjSH6vMsZsJiilY2GDHUrc=s0-d)
dengan demikian dapat disimpulkan bahwa
![y=\sum_{0}^{\infty}a_n x^n](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uQ_LgyzSWEZCs-7k4Oxp9QjbknnGr3AFbKCOIm5F5xY9gb0PjkhOaJscCxMHnkrwVg218M0gprtzFCx50IisGIUEhnESHyCcnymFteNEPqlIQQeLXvqgNAbOop5DQz2RTqsjw5fOxLmH-mnk0QIWdT-hJO0sAK=s0-d)
![y=\sum_{0}^{\infty}-\frac{aa_0}{n!} x^n=-aa_0\sum_{n=0}^{\infty}\frac{x^n}{n!}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v5HCh_9o5Q9F9a1Th0NV5mPE2CO1Oc55tRketMJe7yh6ff1vG9SVPwahtG9rFKqLwsN3ybQeCh6HKDM3_0T7JpTvUgqCsco4tpyMu5UDY0U3zREew0JjrG7tyA3yDh5EwR7XQGxy0bu82_RHLGXdqS-sgFdtGoPioGeug_DLX-JDYjclPw6F3CI2oZiHq_a4o1giy9v41LIuYo746bcqSwQ4Jty18Pb6wwNTkM4iLnuVJpBfHtqh4BVSFhLitr3y2X5VStC09pMdHRqcIschisFQ=s0-d)
dimana
![\sum_{n=0}^{\infty}\frac{x^n}{n!}=e^x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sIm_Eltu8bfIfzdGzZWO1P0RLb-PozkZZbVrtyaRGrPrZQf5XfcmnXlQhXD41qNDqSHTmRO57TFgLfuEwB8U7quBNIfGzwYkOBEgWKGdAqaJdgAL5OeVX8YWPoz91iB1HdSUzYIRMdYzrfASYCqlBGkSDnczqPT-cyc5ZbbTW36feMW0eMtOI0JeQ=s0-d)
sehingga
![y=-aa_0e^x](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_ux9hWiIQNkquG8j7jEFUcKppm6WYbRi7vQed1EBKFxyFkmVxpBBicBi7OK6-xft8tOEytarchPx3m6TDa9gOf4mLRPWt3B6aC7zVgg7dwSldgzDA=s0-d)
adalah juga merupakan solusi dari persamaan diatas.
mudah kan..?????????? yang penting you mengerti tentang turunan, maka yakin bahwa persamaan-persamaan begini cuman yaaaaaaa, kata senior saya CETEK alias gampang....
oh ya, ngomong-ngomong ini salah satu soal FISMAT yang diberikan sebagai Tugas, dan saya selesaikan disini, siapa tahu bermanfaat..
Thanxxx....
Untuk menyelesaikan soal diatas, kita misalkan
miaslkan
karna
sekarang kita hitung untuk
dengan demikian dapat disimpulkan bahwa
dimana
sehingga
adalah juga merupakan solusi dari persamaan diatas.
mudah kan..?????????? yang penting you mengerti tentang turunan, maka yakin bahwa persamaan-persamaan begini cuman yaaaaaaa, kata senior saya CETEK alias gampang....
oh ya, ngomong-ngomong ini salah satu soal FISMAT yang diberikan sebagai Tugas, dan saya selesaikan disini, siapa tahu bermanfaat..
Thanxxx....
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