Examples

Chapter 9 Class 12 Differential Equations
Serial order wise

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Example 14 In a bank, principal increases continuously at the rate of 5% per year. In how many years Rs 1000 double itself ? Rate of change of principal = 5% of Principal ππ/ππ‘ = 5% Γ P ππ/ππ‘ = 5/100 Γ P ππ/ππ‘ = 1/20 Γ P ππ/π = ππ‘/20 Integrating both sides β«1βππ/π = β«1βππ‘/20 log |π| = π‘/20 + C Removing log P = e^(π‘/20 + πΆ) Γ ec P = e^(π‘/20 + πΆ) Γ ec P = ke^(π‘/20) where k = ec Now, we have to find in how many years Rs 1000 double it self Thus, we need to find time T when Principal is Rs 2000 First let us find k At t = 0, P = 1000 Putting in (1) P = ke^(π‘/20) 1000 = ke^(0/20) 1000 = k e0 1000 = k Γ 1 1000 = k So, k = 1000 Put k = 1000 in (1) P = ke^(π‘/20) P = 1000 e^(π‘/20) Now, we need to find time t when Principal is Rs 2000 Putting P = 2000, t = t 2000 = 1000 e^(π‘/20) 2000/1000 = e^(π‘/20) 2 = e^(π‘/20) e^(π‘/20) = 2 Taking log both sides log_πβ‘γπ^(π‘/20) γ=log_πβ‘2 t/20 log_πβ‘π=log_πβ‘2 " " t/20Γ1=log_πβ‘2 π­=ππ γπππγ_πβ‘π (log xπ = πlog x) (log e = 1)