$\frac{d R}{d S} = k (\frac{R}{S})$ $(dR)/(dS) = k (R/S)$ R=?

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Answer 1 · 2018-06-28T17:27:11

$R = C S^{k}$ $R = CS^k$ where C is an arbitrary constant.

Given: $\frac{d R}{d S} = k (\frac{R}{S})$ $(dR)/(dS) = k (R/S)$

Use the separation of variables method:

$\frac{d R}{R} = k \frac{d S}{S}$ $(dR)/R = k (dS)/S$

Integrate both sides:

$\int \frac{d R}{R} = k \int \frac{d S}{S}$ $int(dR)/R = kint (dS)/S$

$ln (R) = k ln (S) + C$ $ln(R) = kln(S)+ C$

$ln (R) = ln (S^{k}) + C$ $ln(R) = ln(S^k)+ C$

$e^{ln (R)} = e^{ln (S^{k}) + C}$ $e^(ln(R)) = e^(ln(S^k)+ C)$

$R = e^{C} e^{ln (S^{k})}$ $R = e^Ce^ln(S^k)$

$R = e^{C} S^{k}$ $R = e^CS^k$

$R = C S^{k}$ $R = CS^k$ where C is an arbitrary constant.

(dR)/(dS) = k (R/S)dRdS=k(RS)(dR)/(dS) = k (R/S) R=?