CAIE A-Level Mathematics 3.8.2 Separable Differential Equations

Practise separating first-order differential equations, integrating both sides and applying an initial condition.

Exam points

move all y terms with dy and all x terms with dx before integrating
use the required integration technique and include an arbitrary constant
apply the initial condition, then rearrange into the requested relation or explicit form

[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

B1

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

Obtain term $\ln x$

B1

Obtain term $-10 k \mathrm{e}^{-0.1 t}$

B1

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 $a b \neq 0$

M1

Obtain $\ln x=10 k\left(1-\mathrm{e}^{-0.1 t}\right)+\ln 20$

A1

or equivalent ISW

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3.8.2—Differential equations question 1