Step 1: Understanding the Question:
We need to find the particular solution of a first-order separable differential equation given an initial condition. Step 3: Detailed Explanation:
1. Start with the given equation: $\log(\frac{dy}{dx}) = 2x - 5y$.
Take exponent on both sides: $\frac{dy}{dx} = e^{2x - 5y} = e^{2x} \cdot e^{-5y}$.
2. Separate the variables:
$\frac{1}{e^{-5y}} dy = e^{2x} dx \implies e^{5y} dy = e^{2x} dx$.
3. Integrate both sides:
$\int e^{5y} dy = \int e^{2x} dx$
$\frac{e^{5y}}{5} = \frac{e^{2x}}{2} + c \implies 2e^{5y} = 5e^{2x} + 10c$.
Let $C = -10c$ be a new constant: $5e^{2x} - 2e^{5y} = C$.
4. Use initial condition $y(0) = 0$:
$5e^{2(0)} - 2e^{5(0)} = C \implies 5(1) - 2(1) = C \implies C = 3$.
5. Final equation: $5e^{2x} - 2e^{5y} = 3$. Step 4: Final Answer:
The solution is $5e^{2x} - 2e^{5y} = 3$.