CAIE A-Level Mathematics A2 3.8.2 Differential Equations Questions

Practise solving separable differential equations using integration and initial conditions.

Exam points

separate variables and integrate differential equations
apply initial conditions to determine constants
interpret solutions and evaluate variables at specified values

[Maximum number: 5]

In a certain chemical reaction the amount, x grams, of a substance is increasing. The differential equation satisfied by x and t, the time in seconds since the reaction began, is

\frac{\mathrm{d} x}{\mathrm{~d} t}=k x \mathrm{e}^{-0.1 t},

where k is a positive constant. It is given that x=20 at the start of the reaction.

Solve the differential equation, obtaining a relation between x, t and k.

Separate variables correctly
$\int \frac{1}{x} \mathrm{~d} x=\int k \mathrm{e}^{-0.1 t} \mathrm{~d} t$

Obtain term $\ln x$
Obtain term $-10 k \mathrm{e}^{-0.1 t}$
Not from $\int x \mathrm{e}^{-0.1 t} \mathrm{~d} t$

Use x=20, t=0 to evaluate a constant or as limits in a solution containing
terms $a \ln x, b e^{-0.1 t}$ where $a b \neq 0$
Obtain $\ln x=10 k\left(1-\mathrm{e}^{-0.1 t}\right)+\ln 20$
or equivalent ISW

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CAIE A-Level Mathematics A2 3.8.2 Differential Equations Questions question 1