text: find the solution of the differential equation y"+16y'=0 satisfying ths conditions y(0)= -5 and y(pi/8)=7 im trying to do math homework but i cant figure out how to do this one question. please help. ok, the characteristic equation is λ² + 16λ + 0 = 0 the roots are λ = 0, -16 the characteristic roots are e^0, e^(-16x)  the solution to the diff. eq. is of the form: ae^(-16x) + b  plug in the initial values: ae^(-16*0) + b = -5 => b = -5 ae^(-16π/8) - 5 = 7 => a = 12/e^(-2π), approximately 6425.9  your final solution is: 12e^(-16x - 2π) - 5