Topic 3 of 6 · Form 6
Further Pure 2: hyperbolic functions, further calculus and differential equations
Definitions of sinh, cosh, tanh; identities (cosh²x − sinh²x = 1); inverses in logarithmic form; derivatives and integrals including ∫1/√(x² ± a²) dx.
Notes for this topic 1 file, in order
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Hyperbolic functions
Definitions of sinh, cosh, tanh; identities (cosh²x − sinh²x = 1); inverses in logarithmic form; derivatives and integrals including ∫1/√(x² ± a²) dx.
Worked example: Solve 3 cosh x − sinh x = 3.
3(eˣ + e⁻ˣ)/2 − (eˣ − e⁻ˣ)/2 = 3 → eˣ + 2e⁻ˣ = 3 → e²ˣ − 3eˣ + 2 = 0 → eˣ = 1 or 2 → x = 0 or ln 2
Further calculus
Differentiation of inverse trigonometric and hyperbolic functions; implicit second derivatives; Maclaurin series; reduction formulae; arc length and surface area of revolution (Cartesian and parametric); limits of areas using rectangles.
Second-order differential equations
ay″ + by′ + cy = f(x): complementary function from the auxiliary equation (real distinct, repeated, complex roots); particular integral for polynomial, exponential and trigonometric f(x); boundary conditions; substitutions to reduce equations.
Worked example: Find the general solution of y″ + 4y′ + 4y = 8.
m = −2 (repeated) → CF (A + Bx)e⁻²ˣ; PI y = 2 → y = (A + Bx)e⁻²ˣ + 2
First-order linear equations
Integrating factor method: dy/dx + P(x)y = Q(x), multiply by e^(∫P dx).
Check yourself
- Express arsinh 2 in logarithmic form.
- Find the integrating factor for dy/dx + (2/x)y = x.
- State the CF of y″ + 9y = 0.
Show answers
- ln(2 + √5)
- x²
- A cos 3x + B sin 3x
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