Explanation Method used: Separation of variables: “Suppose we are given a first-order separable differential equation in the form, d y d x = f ( x ) g ( y ) Step 1: Separate the variables by writing d y d x = f ( x ) g ( y ) in the form, d y g ( y ) = f ( x ) d x Step 2: Integrate each side of d y g ( y ) = f ( x ) d x with respect to the appropriate variable”. Calculation: The given differential equation is d Q d t = k Q ( C − ln Q ) , where k is a positive constants. Separate the variables in the form of d y g ( y ) = f ( x ) d x . d Q Q ( C − ln Q ) = k d t Integrate the above equation with respect to the appropriate variable as follows. ∫ d Q Q ( C − ln Q ) = k ∫ d t − ∫ d u u = k ∫ d t [ Let u = ( C − ln Q ) d u = − 1 Q d Q ] − ln | u | = k t + P − ln | C − ln Q | = k t + P On further simplification, e − ln | C − ln Q | = e k t + P 1 C − ln Q = P e k t [ ∵ e

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Differential Equations

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Chapter 9.3, Problem 25E

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To solve: The differential equation dQdt=kQ(C−lnQ) with the initial condition Q(0)=Q0.

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Chapter 9.3, Problem 23E

Chapter 9.3, Problem 26E

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The differential equation d Q d t = k Q ( C − ln Q ) with the initial condition Q ( 0 ) = Q 0 .

Solution Summary: The author explains how to solve a differential equation with the initial condition Q(0)=Q_0.

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To solve: The differential equation dQdt=kQ(C−lnQ) with the initial condition Q(0)=Q0.

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Chapter 9 Solutions