Answer: \(y=2e^{5x}+3x+7\).

Integrate the gradient:

\(y=\int(10e^{5x}+3)\,dx.\)

This gives

\(y=2e^{5x}+3x+C.\)

The curve passes through \((0,9)\), so

\(9=2e^0+3(0)+C.\)

Thus

\(9=2+C,\)

so

\(C=7.\)

Therefore the equation of the curve is

\(\boxed{y=2e^{5x}+3x+7}.\)