Solving a Logistic IVP

Differential Equations · Axiom Academy

EXAMPLE Solving a Logistic IVP Complete solution using separation of variables and partial fractions Solve the logistic initial value problem with the given growth rate, carrying capacity, and starting population. Excellent work! You've successfully solved a logistic initial value problem. Here's what we learned: Separation of Variables: The logistic equation can be solved by separating variables, collecting all y-terms on one side and t-terms on the other. Partial Fractions: The key step is decomposing into partial fractions, , which makes integration possible — and the factors of K cancel cleanly against the from decomposing, so the exponent stays rt . Logarithmic Integration: After partial fractions, we integrate to get logarithms, then combine and exponentiate to remove them. Initial Condition: The initial condition y(0)=100 determines the constant, giving us the unique solution to our IVP. Logistic Behavior: The final solution shows the characteristic S-curve, starting at 100 and approaching the carrying capacity K=1000 . This technique of separation of variables combined with partial fractions is essential for solving many differential equations in population dynamics, economics, and biological systems!

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