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

Question

anonymous · Answer

can you help me?

sepeario · Answer

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

loser66 · Answer

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.

anonymous · Answer

$$y=c1x+c2 e ^{2x}   ...(1)$$
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)