1) Find the general solution to the following differential equation.

Question

dy/dx+y(cot x)=5e^x

Answer 1

This is not separable, but we can use an integration factor (in this case sin(x)) which we will multiply both sides by to make the left-hand side of the diff eq the recognizable derivative of a product of functions:

Multiply by sin(x):

sin(x)·dy/dx +cos(x)·y = 5e^xsin(x)

d[sin(x)·y]/dx = 5e^xsin(x). We integrate both sides now; the right-hand side is integrable using int by pts:

sin(x)·y = [-5e^xcos(x) +e^xsin(x)] / 2 + C

y = [-5e^xcot(x) + e^x] / 2 + C / sin(x)