Differential Equations

Notes for SL students · bridge page

Is this topic in the SL course?

No. Solving differential equations (separation of variables, slope fields, Euler's method) belongs to the HL-only part of the course. It will not be examined on SL papers.

However, SL students already solve the simplest type of differential equation, written as dy/dx = f(x), whenever they find a function from its derivative and one known point. This page shows that skill, which is examined at SL.

1 What a differential equation is

A differential equation is any equation that contains a derivative, for example dy/dx = 6x². Solving it means finding y as a function of x. Because integrating always produces a constant c, the general solution is a whole family of curves. A known point (an initial condition) picks out one curve: the particular solution.

2 The SL method

Step 1: integrate both sides with respect to x, remembering + c. Step 2: substitute the known point to find c. Step 3: write the final equation for y.

3 Worked examples

Example 1. A curve has gradient function dy/dx = 6x² − 4 and passes through (1, 5). Find the equation of the curve.

y = ∫(6x² − 4) dx = 2x³ − 4x + c. Substituting (1, 5): 5 = 2 − 4 + c, so c = 7. Hence y = 2x³ − 4x + 7.

Example 2. Given f′(x) = e^2x + 1/x for x > 0, and f(1) = 3, find f(x).

f(x) = ½e^2x + ln x + c. Using f(1) = 3: 3 = ½e² + 0 + c, so c = 3 − ½e². Hence f(x) = ½e^2x + ln x + 3 − ½e².

Example 3 (kinematics). A particle has velocity v(t) = 3t² − 2t m s⁻¹. Its displacement at t = 0 is 4 m. Find s(t) and the displacement at t = 2.

s(t) = ∫v dt = t³ − t² + c. At t = 0, s = 4, so c = 4 and s(t) = t³ − t² + 4. At t = 2: s = 8 − 4 + 4 = 8 m.

4 Practice

1. dy/dx = 4x − 3 and the curve passes through (2, 1). Find y.
2. f′(x) = 3 cos x and f(0) = 2. Find f(x).
3. a(t) = 6t m s⁻², v(0) = 1 m s⁻¹. Find v(t).

Answers: 1. y = 2x² − 3x − 1 2. f(x) = 3 sin x + 2 3. v(t) = 3t² + 1

5 Where to revise this

Integration with an initial condition: Integration and Its Applications (sections on indefinite integrals and kinematics). Rates of change: Derivatives and Their Applications.

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