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Perubahan harga suatu komoditas (dalam dolar) mengikuti persamaan

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\$p'(t)=(q^D(t)-q^S(t))-\\dfrac{2}{3}\\displaystyle\\int_0^t (q^S(s)-q^D(s))\\, ds.\$

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Diketahui jumlah penawaran komoditas (dalam unit) mengikuti \$$q^S(t)=20+2p(t)\$$ dan jumlah permintaan mengikuti \$$q^D(t)=56-p(t)\$$.

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Jika fungsi \$$p'\$$ diturunkan sekali lagi, akan diperoleh persamaan diferensial orde dua berbentuk \$$p''+ap'+bp=k\$$. Nilai dari \$$a\$$, \$$b\$$, dan \$$k\$$ berturut-turut adalah $\\ldots$.

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Catatan: Ingat Teorema Dasar Kalkulus, $\\dfrac{d}{dt}\\displaystyle\\int_0^t f(s)\\, ds=f(t)$.

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Solusi dari persamaan differensial orde $2$ sebelumnya adalah \$$p_h(t)=C_1e^{\\alpha t}+C_2e^{\\beta t}+\\hat p\$$. Maka \$$\\alpha+\\beta+\\hat p=\\ldots\$$.

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Jika diberikan syarat awal $p(0)=0$ dan $p'(0)=0$, tentukan nilai \$$C_1\\cdot C_2\$$.

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