Integration (A2 Unit 4: Applied Mathematics B) — WJEC A-Level Mathematics
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Integration (A2 Unit 4: Applied Mathematics B) explained
First order differential equations of the form dy/dx = f(x)g(y) are solved using separation of variables: rewrite as ∫ (1/g(y)) dy = ∫ f(x) dx.
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Performing the integration produces a general solution containing an arbitrary constant of integration C. A particular solution is found by substituting given initial or boundary conditions (such as y = y₀ when x = x₀) to determine the unique numerical value of C. Candidates must manipulate the resulting equation algebraically to express y explicitly in terms of x where required, handling logarithmic and exponential terms fluently.
Your focus
- Separate variables to solve first-order differential equations analytically.
- Evaluate the arbitrary constant of integration using given initial boundary conditions.
- Express particular solutions explicitly in terms of the independent variable.
Integration (A2 Unit 4: Applied Mathematics B) exam tips
Marking Points
- separating variables correctly with integrals on both sides: ∫ (1/g(y)) dy = ∫ f(x) dx
- integrating both sides correctly, including an arbitrary constant of integration + C
- substituting initial boundary conditions to evaluate the constant of integration C
- rearranging the solution algebraically to express y explicitly in terms of x
Examiner Tips
- 💡Include '+ C' immediately when you integrate: ln|y| = kx + C leads to y = A e^(kx) where A = e^C.
- 💡Substitute initial conditions as early as possible to find C before complicated algebraic rearrangements.
Common Mistakes
- adding the constant of integration C at the very end after exponentiating, rather than immediately upon integration
- failing to separate variables properly before attempting integration