Differential Equations
Linear Differential Equations
Grade 12
Question:
<p>Let the population of rabbits surviving at a time \(t\) be governed by the differential equation \(\dfrac{dp(t)}{dt} = \dfrac{1}{2}p(t) - 200\). If \(p(0) = 100\), then \(p(t)\) equals</p>
<p>\(600 - 500\, e^{t/2}\)</p>
<p>\(400 - 300\, e^{-t/2}\)</p>
<p>\(400 - 300\, e^{t/2}\)</p>
<p>\(300 - 200\, e^{-t/2}\)</p>
Step-by-Step Solution
Key Concept: This is a first-order linear differential equation of the form dp/dt = ap + b. Solve it by finding the equilibrium point, then use the homogeneous solution with initial conditions to get the complete solution.
<p><strong>Step 1: Rewrite in standard form</strong></p><p>dp/dt - (1/2)p = -200</p><p><strong>Step 2: Find equilibrium (particular solution)</strong></p><p>At equilibrium, dp/dt = 0: 0 = (1/2)p - 200 ⟹ p_eq = 400</p><p><strong>Step 3: Solve homogeneous equation</strong></p><p>dp/dt - (1/2)p = 0 gives p_h = Ce^(t/2)</p><p><strong>Step 4: General solution</strong></p><p>p(t) = 400 + Ce^(t/2)</p><p><strong>Step 5: Apply initial condition p(0) = 100</strong></p><p>100 = 400 + C ⟹ C = -300</p><p><strong>Step 6: Final solution</strong></p><p>p(t) = 400 - 300e^(t/2)</p><p>∴ Answer: C</p>
Correct Answer: C