Question

Solve the differential equation y' = cos(x + y) with the initial condition y (0)= π / 2 . Select the correct solution.

a)y = sin⁻¹(x + 1) - x
The method of the section cannot be applied since there is no constant with (x + y)
y = ln(sec x + tan x) + π / 2
y = 2 tan⁻¹(x + 1) - x

Answer 1

The differential equation y' = cos(x + y) with the initial condition y(0) = π/2 can be solved by separating the variables and integrating. The correct solution is y = ln|sec(y) + tan(y)| - x.

Step-by-step explanation:

The given differential equation is y' = cos(x + y) with the initial condition y(0) = π/2.

To solve this, we can separate the variables.

Rearranging the equation, we get:

dy/cos(y) = dx

Integrating both sides, we have:

∫dy/cos(y) = ∫dx

Applying the inverse trigonometric identity, we get:

ln|sec(y) + tan(y)| = x + C

Substituting the initial condition, we find that C = ln(1 + tan(π/2)) = ln(1 + ∞) = ∞.

Therefore, the correct solution is y = ln|sec(y) + tan(y)| - x, with the initial condition y(0) = π/2.