Find the general solution of the differential equation `(dy)/(dx) - 2y = cos 3x.`.

Explanation

The given differential equation is of the form ` (dy)/(dx) + Py = Q`, where `P = - 2 and Q = cos 3x `.
Thus, the given differential equation is linear.
` IF= e ^( int Pdx) = e ^(int - 2dx) = e ^( - 2x )`
So, the required solution is
`y xx IF = int {Q xx IF } dx +C`,
i.e., `y xx e ^( - 2x ) = int e ^(- 2x ) cos 3x dx +C`
` " " = e ^(-2x ) [ ( -2 cos 3x + 3 sin 3x)/({ (-2) ^(2) + 3^(2)}) ] + C `
` " " [ because int e ^(ax) cos bx dx = e ^( ax) {( a cosbx + b sin bx )/( (a ^(2) + b ^(2))}]`
` therefore y = (( 3 sin 3x - 2 cos 3x ))/(13) + Ce^( 2x )`, which is the required solution.