What is the general solution to this differential equation

y' = -0.5y + t
y(1) = 6
It will be helpful if you show me your work. Thanks!

You need to find the integrating factor µ(t) to solve this equation. You multiply this by both sides and solve for y(t)

µ(t) - exp ∫0.5dt = e^0.5t

So y(t) = 1/µ(t) * ∫µ(t)*t dt = e^-0.5t*e^0.5t(2t-4) +ce-0.5t => y = 2t - 4 + ce^-0.5t

Using y(1)=6 you have 6 = 2 - 4 + ce^-0.5 or c = 8*e^0.5

So y(t) = 2t + 8e^0.5*e^-0.5t - 4

Hope that helps!