Question

eliminate the arbitrary constant y=c1x=c2e^2x

Answer 1

can you help me?

Answer 2

can you please write your question in correct notation? it's really hard to read it.

Answer 3

I guess!! it is \(y = C_1x+C_2e^{2x}\) and it is a solution of a DE. If it is so, to eliminate the constants, you need initial values, like y(0) = a, and y'(0) =b.

Answer 4

\[y=c1x+c2 e ^{2x} ...(1)\]
diff. twice w.r.t x
\[y \prime=C1+2 C2 e ^{2x} ...(2)\]
\[y \prime \prime=0+4C2 e ^{2x} ...(3)\]
from (3)
\[C2=\frac{ y \prime \prime }{ 4~e ^{2x} }\]
put in (2)
\[y \prime=C1+2 e ^{2x}*\frac{ y \prime \prime }{ 4~e ^{2x} }\]
\[C1=y \prime-\frac{ y \prime \prime }{ 2 }\]
put the values of C1 and C2 in (1)