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Diberikan sistem persamaan diferensial \$$\\mathbf{x}'=A\\mathbf{x}\$$ dengan \$$A=\\begin{bmatrix} \\var{a} & \\var{-a} \\\\ \\var{a} & \\var{a} \\end{bmatrix}\$$.

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Jenis dan sifat kestabilan solusi kesetimbangan sistem tersebut di titik \$$(0,0)\$$ adalah \$$\\ldots\$$.

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Diberikan masalah nilai awal \$$\\mathbf{x}'=\\begin{bmatrix} -2 & 1 \\\\ -5 & 4 \\end{bmatrix}\\mathbf{x}\$$, \$$\\mathbf{x}_0=\\begin{bmatrix} 1 \\\\ 5 \\end{bmatrix}\$$.

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Jika solusi khusus masalah nilai awal tersebut adalah \$$\\mathbf{x}(t)=\\begin{bmatrix} ae^{bt} \\\\ ce^{bt} \\end{bmatrix}\$$, maka \$$a+b+c=\\ldots\$$.

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Tentukan pasangan sistem persamaan diferensial berikut dan jenis kestabilannya di titik \$$(0,0)\$$ yang tepat.

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Diberikan masalah nilai awal $\\begin{bmatrix} x_1'(t) \\\\ x_2'(t) \\end{bmatrix}=\\begin{bmatrix} -\\var{2*a} & \\var{a} \\\\ \\var{2*a} & -\\var{a} \\end{bmatrix}\\begin{bmatrix} x_1(t) \\\\ x_2(t) \\end{bmatrix}$, $\\begin{bmatrix} x_1(0) \\\\ x_2(0) \\end{bmatrix}=\\begin{bmatrix} \\var{b} \\\\ \\var{b} \\end{bmatrix}$.

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$x_1(\\var{c})+x_2(\\var{c})=\\ldots$.

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Petunjuk: Tanpa menentukan solusi umumnya, hitung $\\dfrac{d}{dt}(x_1(t)+x_2(t))$ terlebih dahulu.

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Hemodialisis adalah proses yang dilakukan oleh suatu mesin untuk menyaring urea dari darah pasien jika ginjal pasien bermasalah. Banyaknya urea saat dialisis seringkali dimodelkan dengan memisalkan terdapatnya dua buah tempat tersimpannya urea: pada darah pasien, yang secara langsung disaring oleh mesin dialisis, dan tempat lain yang tidak bisa disaring secara langsung oleh mesin dialisis. Sistem persamaan diferensial yang mendeskripsikan model tersebut adalah
\$\\begin{cases} \\dfrac{dc}{dt} &= -\\dfrac{K}{V}c+ap-bc \\\\ \\dfrac{dp}{dt} &= -ap+bc \\end{cases}\$
dengan \$$c\$$ dan \$$p\$$ adalah konsentrasi urea di darah pasien dan di tempat lain (dalam mg/mL) dan semua konstanta lainnya bernilai positif. Misal \$$K=3\$$, \$$V=2\$$, \$$a=b=1\$$, konsentrasi urea awal \$$c(0)=5\$$ mg/mL, dan \$$p(0)=5\$$ mg/mL.

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Jika \$$\\mathbf{x}(t)=\\begin{bmatrix} c(t) \\\\ p(t) \\end{bmatrix}\$$, tentukan matriks \$$A\$$ agar model dapat dituliskan menjadi masalah nilai awal \$$\\mathbf{x}'=A\\mathbf{x}\$$, \$$\\mathbf{x}(0)=\\begin{bmatrix} c_0 \\\\ p_0 \\end{bmatrix}\$$.

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Diketahui

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• Matriks \$$A\$$ mempunyai dua buah nilai eigen berbeda, \$$\\lambda_1\$$ dan \$$\\lambda_2\$$, dengan \$$\\lambda_1>\\lambda_2\$$.
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• Vektor eigen yang berkorespondensi dengan nilai eigen \$$\\lambda_1\$$ adalah \$$\\mathbf{v}_1=\\begin{bmatrix} a \\\\ 2 \\end{bmatrix}\$$.
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• Vektor eigen yang berkorespondensi dengan nilai eigen \$$\\lambda_2\$$ adalah \$$\\mathbf{v}_2=\\begin{bmatrix} b \\\\ 1 \\end{bmatrix}\$$.
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Tentukan nilai $\\lambda_1$, $\\lambda_2$, $a$, dan $b$ berturut-turut.

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Solusi umum dari model tersebut dapat dituliskan menjadi $\\mathbf{x}(t)=C_1\\mathbf{v}_1e^{\\lambda_1t} + C_2\\mathbf{v}_2e^{\\lambda_2t}$.

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Tentukan $C_1$ dan $C_2$.

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Tentukan \$$\\lim\\limits_{t\\to\\infty}c(t)\$$.

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Waktunya 5 menit lagi ya.

Selamat datang!

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Silakan gunakan set soal ini dengan bijak. Kalian bisa mencoba set soal ini terus-menerus.

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Set soal ini dapat digunakan oleh siapapun secara gratis.

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Petunjuk pengerjaan soal:

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• Jawaban dapat dituliskan dalam bentuk desimal maupun pecahan.
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• Jika jawaban dituliskan dalam bentuk desimal, pisahkan dengan tanda titik dan bulatkan hingga dua angka di belakang titik. Contoh: ketik 0.67 untuk menjawab \$$\\frac{2}{3}\$$.
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• Jika jawaban dituliskan dalam bentuk pecahan, pisahkan pembilang dan penyebut dengan garis miring dan jawab dengan pecahan paling sederhana. Contoh: ketik 2/3 untuk menjawab \$$\\frac{2}{3}\$$.
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Jika ada kunci jawaban yang salah, kalian dapat memberitahu kami di Instagram @meongmeongproject atau OA Line @eog7710d.

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