The differential equation is and the function that is to be proven as solution is

y = be^x+ ce^2x, now we need to find the values of and .

be^x + 2ce^2x

be^x + 4ce^2x

Putting the values of these variables in the differential equation, we get,

be^x + 4ce^2x – 3(be^x + 2ce^2x) + 2(be^x + ce^2x) = 0,

0 = 0

As, L.H.S = R.H.S. the equation is satisfied. Hence, this function is the solution of the differential equation.