The variables $$\(x\)$$ and $$\(t\)$$ satisfy the differential equation $$\(\frac{\mathrm{d} x}{\mathrm{~d} t}=x^{2}(1+2 x)\)$$, and $$\(x=1\)$$ when $$\(t=0\)$$. Using partial fractions, solve the differential equation, obtaining an expression for $$\(t\)$$ in terms of $$\(x\)$$. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
Exam No：9709_s21_qp_31 Year：2021 Question No：10
Answer:
State a suitable form of partial fractions for \(\frac{1}{x^{2}(1+2 x)}\)
Use a relevant method to determine a constant
Obtain one of \(A=-2, B=1\) and \(C=4\)
Obtain a second value
Obtain the third value
Separate variables correctly and integrate at least one term
Obtain terms \(-2 \ln x-\frac{1}{x}+2 \ln (1+2 x)\) and \(t\)
Evaluate a constant, or use limits \(x=1, t=0\) in a solution containing terms \(t\), \(a \ln x\)
and \(b \ln (1+2 x)\), where \(a b \neq 0\)
Obtain a correct expression for \(t\) in any form, e.g. \(t=-\frac{1}{x}+2 \ln \left(\frac{1+2 x}{3 x}\right)+1\)
Use a relevant method to determine a constant
Obtain one of \(A=-2, B=1\) and \(C=4\)
Obtain a second value
Obtain the third value
Separate variables correctly and integrate at least one term
Obtain terms \(-2 \ln x-\frac{1}{x}+2 \ln (1+2 x)\) and \(t\)
Evaluate a constant, or use limits \(x=1, t=0\) in a solution containing terms \(t\), \(a \ln x\)
and \(b \ln (1+2 x)\), where \(a b \neq 0\)
Obtain a correct expression for \(t\) in any form, e.g. \(t=-\frac{1}{x}+2 \ln \left(\frac{1+2 x}{3 x}\right)+1\)
Knowledge points:
3.2.1 understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base)
3.5.3 integrate rational functions by means of decomposition into partial fractions (Restricted to types of partial fractions as specified in topic 3.1 above.)
3.8.2 find by integration a general form of solution for a first order differential equation in which the variables are separable (Including any of the integration techniques from topic 3.5 above.)
3.8.3 use an initial condition to find a particular solution
Solution:
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