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📐 Mathematics  ·  Differential Equations  ·  JEE

Population growth modeled as dP/dt = kP, with P(0) = 1000, P(1) = 1500. Find k:

Answer: ln(1.5).

  • A ln(1.5)
  • B 0.5
  • C ln(2)
  • D 1.5

Correct answer: A. ln(1.5)

Explanation: P = 1000 e<sup>kt</sup>. At t=1: 1500 = 1000 e<sup>k.</sup> k = ln(1.5).

Family of Solution Curves: y = Ax²Each value of the constant A gives ONE particular curve from the family

The general solution y=Ax² represents an entire FAMILY of curves, one for each value of the arbitrary constant A; a particular solution (fixed by an initial condition) selects exactly one curve from this family.

Concept context

Equations involving derivatives and their solutions

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