Question

Find the particular solution of the differential equation \frac{dy}{dx} + y\cos(x) = 5\cos(x)

Answer 1

`(\mathrm dy)/(\mathrm dx)+y\cos x=5\cos x`

`e^(\sin x)(\mathrm dy)/(\mathrm dx)+y\cos xe^(\sin x)=5\cos xe^(\sin x)`

`(\mathrm d)/(\mathrm dx)\left[e^(\sin x)y\right]=5\cos xe^(\sin x)`

`e^(\sin x)y=5\displaystyle\int\cos xe^(\sin x)\,\mathrm dx`

`e^(\sin x)y=5e^(\sin x)+C`

`y=5+Ce^(-\sin x)`