Chapter 10 - Introduction to Differential Equations - 10.1 Solving Differential Equations - Exercises - Page 505: 31

$$ y =75 e^{-2 x} $$

Given $$ y^{\prime} =-2 y,\ \ \ y( \ln 5) =3 $$ Then \begin{aligned} y^{\prime} &=-2 y \\ \frac{d y}{d x} &=-2 y \\ \frac{d y}{y} &=-2 d x \\ \ln y &=-2 x+C \end{aligned} Since $y( \ln 5) =3$, then $\ln 3= -2\ln 5+C \ \to C= \ln 75 $, then \begin{aligned} \ln \frac{y}{75} &=-2 x \\ \frac{y}{75} &=e^{-2 x} \\ y &=75 e^{-2 x} \end{aligned}