Math Problem Statement

MathBio-HUEB-11.pdf

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Solve the excercises.

Solution

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text[:1000]The document contains exercises related to mathematics for biosciences, including differential equations. Here's a summary of the tasks outlined in the first few pages:

Exercise 1:

Solve the initial value problem for y(t)y(t), where:

  • k,S>0k, S > 0 and 0<y0<S0 < y_0 < S,
  • The differential equation is y(t)=ky(t)ln(Sy(t))y'(t) = k y(t) \ln\left(\frac{S}{y(t)}\right),
  • The initial condition is y(0)=y0y(0) = y_0,
  • Find the solution and the maximum interval on which the solution is defined.

Exercise 2:

Solve the following differential equations and find the maximal interval for each:

  1. y(t)=et+4t3ey(t)y'(t) = e^t + 4t^3 e^{y(t)}, with initial condition y(0)=0y(0) = 0,
  2. y(t)=(2y+3)(4t+5)2y'(t) = \left( 2y + 3 \right) \left( 4t + 5 \right)^2, with initial condition y(0)=23y(0) = \frac{2}{3}.

Exercise 3 (Additional):

Solve the SIR model for epidemic spread, given a population of size NN.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Initial Value Problems
SIR Model

Formulas

y'(t) = k y(t) ln(S/y(t))
y'(t) = e^t + 4t^3 e^{y(t)}
y'(t) = (2y + 3) (4t + 5)^2

Theorems

Solution of Initial Value Problems
Maximal Interval of Solution

Suitable Grade Level

Undergraduate (Biosciences, Mathematics)