Question

eliminate the arbitrary constant

y=C1x+C2e^2x

Answer 1

the arbitrary constant is 1. since anything multiplied by one is itself

Answer 2

Can you please show a solution...

Answer 3

\(y = C_1x + C_2e^{2x}\\y'= C_1+2C_2e^{2x}\rightarrow C_1= y'-2C_2e^{2x}\)
Replace to y
\(y = (y'-2C_2e^{2x})x+ C_2e^{2x}= y'x -C_2e^{2x}(x-1)\)
but \(y"= 4C_2 e^{2x}\rightarrow C_2e^{2x}=\dfrac{1}{4}y"\)
Replace back
\(y= y'x-\dfrac{1}{4}y"(x-1)\\(x-1)y"-4xy'-4y=0\)