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solve-Mdc-dt-P-km-Cm-where-M-P-m-and-k-are-constant-solve-the-equation-given-C-a-when-t-o-

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Question Number 16492 by Sai dadon. last updated on 23/Jun/17
solve Mdc/dt=P+km−Cm where M.P.m and k are constant solve the equation given C=a when t=o
$${solve} \\ $$$${Mdc}/{dt}={P}+{km}−{Cm}\:{where}\:{M}.{P}.{m}\:{and} \\ $$$${k}\:{are}\:{constant} \\ $$$${solve}\:{the}\:{equation}\:{given}\:{C}={a}\:{when}\:{t}={o} \\ $$
Answered by mrW1 last updated on 23/Jun/17
(dc/(P+km−mc))=(dt/M) ∫(dc/(P+km−mc))=∫(dt/M) −(1/m)ln (P+km−mc)=(t/M)+U_1 −Mln (P+km−mc)=mt+U_2 ln (P+km−mc)=−(m/M)t+U_3 at t=0: c=a ln (P+km−ma)=U_3 ln (P+km−mc)−ln (P+km−ma)=−(m/M)t ln ((P+km−mc)/(P+km−ma))=−(m/M)t ((P+km−mc)/(P+km−ma))=e^(−(m/M)t) P+km−mc=(P+km−ma)e^(−(m/M)t) ⇒c=(P/m)+k−((P/m)+k−a)e^(−(m/M)t)
$$\frac{\mathrm{dc}}{\mathrm{P}+\mathrm{km}−\mathrm{mc}}=\frac{\mathrm{dt}}{\mathrm{M}} \\ $$$$\int\frac{\mathrm{dc}}{\mathrm{P}+\mathrm{km}−\mathrm{mc}}=\int\frac{\mathrm{dt}}{\mathrm{M}} \\ $$$$−\frac{\mathrm{1}}{\mathrm{m}}\mathrm{ln}\:\left(\mathrm{P}+\mathrm{km}−\mathrm{mc}\right)=\frac{\mathrm{t}}{\mathrm{M}}+\mathrm{U}_{\mathrm{1}} \\ $$$$−\mathrm{Mln}\:\left(\mathrm{P}+\mathrm{km}−\mathrm{mc}\right)=\mathrm{mt}+\mathrm{U}_{\mathrm{2}} \\ $$$$\mathrm{ln}\:\left(\mathrm{P}+\mathrm{km}−\mathrm{mc}\right)=−\frac{\mathrm{m}}{\mathrm{M}}\mathrm{t}+\mathrm{U}_{\mathrm{3}} \\ $$$$\mathrm{at}\:\mathrm{t}=\mathrm{0}:\:\mathrm{c}=\mathrm{a}\: \\ $$$$\mathrm{ln}\:\left(\mathrm{P}+\mathrm{km}−\mathrm{ma}\right)=\mathrm{U}_{\mathrm{3}} \\ $$$$\mathrm{ln}\:\left(\mathrm{P}+\mathrm{km}−\mathrm{mc}\right)−\mathrm{ln}\:\left(\mathrm{P}+\mathrm{km}−\mathrm{ma}\right)=−\frac{\mathrm{m}}{\mathrm{M}}\mathrm{t} \\ $$$$\mathrm{ln}\:\frac{\mathrm{P}+\mathrm{km}−\mathrm{mc}}{\mathrm{P}+\mathrm{km}−\mathrm{ma}}=−\frac{\mathrm{m}}{\mathrm{M}}\mathrm{t} \\ $$$$\frac{\mathrm{P}+\mathrm{km}−\mathrm{mc}}{\mathrm{P}+\mathrm{km}−\mathrm{ma}}=\mathrm{e}^{−\frac{\mathrm{m}}{\mathrm{M}}\mathrm{t}} \\ $$$$\mathrm{P}+\mathrm{km}−\mathrm{mc}=\left(\mathrm{P}+\mathrm{km}−\mathrm{ma}\right)\mathrm{e}^{−\frac{\mathrm{m}}{\mathrm{M}}\mathrm{t}} \\ $$$$\Rightarrow\mathrm{c}=\frac{\mathrm{P}}{\mathrm{m}}+\mathrm{k}−\left(\frac{\mathrm{P}}{\mathrm{m}}+\mathrm{k}−\mathrm{a}\right)\mathrm{e}^{−\frac{\mathrm{m}}{\mathrm{M}}\mathrm{t}} \\ $$
Commented by Sai dadon. last updated on 23/Jun/17
Thank sr
$${Thank}\:\:{sr} \\ $$

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