(a) Find the solution of the initial-value problem?

dy/dx = cos(4x) / 3 − sin(4x), y = 6 when x = 0.

(b) (i) Find, in implicit form, the general solution of the differential equation
dy/dx = −e^x + e^−x / y(e^x − e^−x + 1)^3 (y > 0).

(ii) Find the corresponding explicit form of this general solution.

(iii) Find the corresponding particular solution that satisfies the initial condition y = 1 when x = 0.

(iv) What is the value of y given by this particular solution when x = 0.8?
Give your answer to 4 decimal places

Solution Preview

(a)

dy/dx = cos(4x) / 3 − sin(4x),
=>y = (1/4)sin(4x)/3 + (1/4)cos(4x) + C
y = 6 when x = 0, => 1/4 +C = 6 => C = 23/4
=>y = (1/12)sin(4x) + ...

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