Answer to Question #297219 in Differential Equations for Sree

Question

Y=c1e^x+c2e^2x y"-2y+3y=0

Answer 1

Solution:

To check whether $y=c_1e^x+c_2e^{2x}$ is the solution of $y''-2y+3y=0$...(i)

Now, $y=c_1e^x+c_2e^{2x}$

$y'=c_1e^x+2c_2e^{2x} \\ y''=c_1e^x+4c_2e^{2x}$

Put these in LHS of (i),

$=y''-2y+3y \\=c_1e^x+4c_2e^{2x}-2(c_1e^x+2c_2e^{2x})+3(c_1e^x+c_2e^{2x}) \\=2c_1e^x+3c_2e^{2x} \\\ne0$

So, it is not the solution of given D.E.