Find the solution of the differential equation that satisfies the given initial condition. y' tan x = 5a + y, y(π/3) = 5a, 0 < x < π/2, where a is a constant.

Note that the above equation is a first order non-homogeneous linear differential equation which is of the form



where (in the problem):



We can solve this equation by multiplying both sides by the integrating factor I(x) which is calculated as 



Multiplying this integrating factor to both sides of equation (1), we have



Note that 



Using this information for equation (2), 



Now, we evaluate the integral on the left side. Let  so that 



Using equation (4), equation (3) becomes



Since , equation (5) becomes



Using this value of the constant, therefore