Find a general solution to the differential equation given below. Primes denote derivatives with respect to $x$. \[ 8 y^{\prime \prime}+10 y^{\prime}=0 \] The general solution of the differential equation is $y(x)=$

Step 1 ：We are given the homogeneous second order linear differential equation with constant coefficients: \(8 y^{\prime \prime}+10 y^{\prime}=0\).

Step 2 ：The general form of such an equation is \(a y'' + b y' + c y = 0\). The general solution to such an equation is given by \(y(x) = C_1 e^{m_1 x} + C_2 e^{m_2 x}\), where \(m_1\) and \(m_2\) are the roots of the characteristic equation \(am^2 + bm + c = 0\) and \(C_1\) and \(C_2\) are arbitrary constants.

Step 3 ：In this case, \(a = 8\), \(b = 10\), and \(c = 0\). So the characteristic equation is \(8m^2 + 10m = 0\).

Step 4 ：The roots of the characteristic equation are \(m_1 = 0\) and \(m_2 = -\frac{5}{4}\).

Step 5 ：Therefore, the general solution to the differential equation is \(y(x) = C_1 e^{0x} + C_2 e^{-\frac{5}{4}x} = C_1 + C_2 e^{-\frac{5}{4}x}\).

Step 6 ：\(\boxed{\text{Final Answer: The general solution to the differential equation is } y(x) = C_1 + C_2 e^{-\frac{5}{4}x}}\)