Give the alternative corresponding to the solution of the equation y' = 5y : x - 5y = c. ln⁡|x| + y = c. 5x - ln⁡|y| = c. x - ln⁡|1/y| = c.

Step-by-step solution:

1. Start with the differential equation y' = 5y .

2. This is a first-order linear differential equation that can be solved by separation of variables.

3. Rearrange the equation: \frac{dy}{y} = 5 dx .

4. Integrate both sides:

\int \frac{1}{y} dy = \int 5 dx

\ln|y| = 5x + C

5. Solve for y :

y = e^{5x + C} = Ce^{5x}

6. Rewrite the constant term to match the alternative forms:

\ln|y| = 5x + \ln|C|

7. Recognize that \ln|C| can be absorbed by a constant term, so:

5x - \ln|y| = C \quad \text{or} \quad \ln|y| = 5x + C'

The corresponding form is:

5x - \ln|y| = c .