JEE Mains · Maths · STD 12 - 9. differential equations
Let the population ofrabbits surviving at time \(t\) be governed by the differential equation \(\frac{{dp\left( t \right)}}{{dt}} = \frac{1}{2}p\left( t \right) - 200\) . If \( p(0)=100 \) ,then \(p(t)\) equals :
- A \(600 - 500{e^{\frac{t}{2}}}\;\)
- B \(\;400 + 300{e^{\frac{t}{2}}}\)
- C \(\;400 - 300{e^{\frac{t}{2}}}\)
- D \(\;300 - 200{e^{\frac{t}{2}}}\)
Answer & Solution
Correct Answer
(C) \(\;400 - 300{e^{\frac{t}{2}}}\)
Step-by-step Solution
Detailed explanation
\(\int \frac{2 d p(t)}{p(t)-400}=\int d t\) \(2 \log |p(t)-400|=t+c \quad \ldots(1)\) \(t=0, p=100\) \(2 \log (300)=c\) From ( 1) \(2 \log |p(t)-400|=t+2 \log (300)\) \(|p(t)-400|=e^{t / 2} \cdot e^{\log (300)}\) \(\boxed{p(t) = 400 - {e^{t/2}}(300)}\)
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