Maths Pure 3 · Differential equations. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
A differential equation is an equation that contains a derivative such as dy/dx. It describes how fast something changes. In this topic you first build the equation from words. The phrase rate of change means a derivative with respect to time, and proportional to means equals a constant k times.
You then solve it by separating the variables: move everything with y to the dy side and everything with x to the dx side, then integrate both sides. This gives the general solution, which contains a constant c.
A given condition, such as y = 3 when x = 0, fixes the value of c and gives the particular solution. You may then be asked what the solution means, for example what happens as t becomes large.
Students add the constant at the very end, for example going from ln y = x2 to y = ex2 + c. The constant must go in at the integration step, so the correct result is y = Aex2, where A = ec.
Answer: y = ∛(3x2 + 27)
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