Answer
$$y =C e^{-\frac{5}{2} x}+\frac{4}{5} $$
Work Step by Step
Given
$$ 2y'+5 y -4=0 $$
\begin{aligned} 2 \frac{d y}{d x}+5 y &=4 \\ 2 \frac{d y}{d x}+5 y-5 y &=4-5 y \\ 2 \frac{d y}{d x} &=4-5 y \\ \frac{2}{4-5 y} \frac{d y}{d x} &=1\\
\int \frac{2}{4-5 y}dy&= \int dx\\
\frac{-2}{5}\ln |4-5y|&=x+C
\end{aligned}
Hence
\begin{aligned} \ln (4-5 y) &=-\frac{5}{2} x-\frac{5}{2} c \\ y &=C e^{-\frac{5}{2} x}+\frac{4}{5} \end{aligned}
