Question
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.
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Answer to a math question 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.
Birdie
4.5103 Answers
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|
5x - \ln|y| = C \quad \text{or} \quad \ln|y| = 5x + C'
The corresponding form is:
5x - \ln|y| = c
1. Start with the differential equation
2. This is a first-order linear differential equation that can be solved by separation of variables.
3. Rearrange the equation:
4. Integrate both sides:
5. Solve for
6. Rewrite the constant term to match the alternative forms:
7. Recognize that
The corresponding form is:
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