Differential Equation dx/dt = k(a-x)(b-x)

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SUMMARY

The differential equation dx/dt = k(a-x)(b-x) with the initial condition x(0)=0 requires integration using partial fractions for proper solution. The correct approach involves rewriting the integral dx/((a-x)(b-x)) and applying partial fraction decomposition. The user encountered an issue with ln(0) when substituting the initial condition, indicating a misstep in the integration process. The solution must avoid incorrect assumptions about the integration of rational functions.

PREREQUISITES
  • Understanding of differential equations, specifically first-order separable equations.
  • Familiarity with integration techniques, particularly partial fraction decomposition.
  • Knowledge of logarithmic functions and their properties.
  • Basic algebraic manipulation skills to handle rational expressions.
NEXT STEPS
  • Study the method of partial fractions in detail, focusing on its application in integration.
  • Learn about the properties of logarithmic functions to avoid pitfalls like ln(0).
  • Practice solving first-order differential equations using separation of variables.
  • Explore examples of integrating rational functions to solidify understanding of the integration process.
USEFUL FOR

Students studying differential equations, mathematics educators, and anyone seeking to improve their integration skills in calculus.

mjbc
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Homework Statement



dx/dt = k(a-x)(b-x)

x(0)=0

The Attempt at a Solution



dx/((a-x)(b-x)) = kdt

Integral (dx/(ab-bx-ax+x^2)) = kt +C

x/ab - (1/b)ln|x| - (1/a)ln|x| - (x^-1) = kt + C

however, if i try to substitute x(0) = 0, i get the ln 0

please help i have no idea if i am doing this right
 
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I don't think the integration dx is correct at all. Use partial fractions.
 
mjbc said:

Homework Statement



dx/dt = k(a-x)(b-x)

x(0)=0

The Attempt at a Solution



dx/((a-x)(b-x)) = kdt

Integral (dx/(ab-bx-ax+x^2)) = kt +C

x/ab - (1/b)ln|x| - (1/a)ln|x| - (x^-1) = kt + C

however, if i try to substitute x(0) = 0, i get the ln 0

please help i have no idea if i am doing this right

NO, 1/(a+ bx+ cx2) is NOT equal to 1/a+ 1/bx+ 1/cx2.

Rule of thumb: since it is harder to factor than to multiply, NEVER multiply terms that are already factored! (Well, almost never.)

Integrate dx/((a-x)(b-x))= dx/((x-a)(x-b) using partial fractions.
 

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