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An explicit solution is one in which the dependent variable is expressed directly in terms of the independent variable and constants.

Let the given function be \(y'' + y = tanx\).

Then, the first derivative of the function is,

\(y' = - (cos x)sec x + sin x ln(sec x + tan x)\)

Using the product rule, the second derivative of the function is,

\(\begin{aligned}{l}y'' &= ( - cos x(sec x tan x) + sin x sec x) + (sin x sec x + cos x ln(sec x + tan x))\\y'' &= - cos x(sec x tan x) + cos x ln( sec x + tan x) + 2sin x sec x\end{aligned}\)

Substitute \(y\), \(y'\) and \(y''\) into the left-hand side of the differential equation.

\(\begin{aligned}{c} - cos x(sec x tan x) + cos xln(sec x + tan x) + 2sin x sec x + ( - (cos x)ln(sec x + tan x)) &= tan x\\ - cos x(sec x tan x) + 2sin x sec x &= tan x\\ - \frac{1}{{sec x}}(sec x tan x) + 2sin x\frac{1}{{cos x}} &= tan x\\ - tan x + 2tan x &= tan x\\tan x &= tan x\end{aligned}\)

That is same as the right-hand side of the differential equation for every real number \(x\). Thus, the indicated function is an explicit solution of the given differential equation.