How to solve
The equation y' = ky says the rate of change is proportional to the current value. Separating and integrating produces exponential solutions y = Ce^(kx), where C absorbs the constant of integration (and the sign of y).
Steps
- Separate: dy/y = k dx.
- Integrate both sides: ln|y| = kx + C₁.
- Exponentiate and absorb constants: y = Ce^(kx).
Example
Find the general solution of y' = 3y.
- dy/y = 3 dx.
- ln|y| = 3x + C₁ ⇒ y = Ce^(3x).
Answer: Ce^(3x)
How to type answers
- Enter like Ce^(3x) or Ce^(-2x)
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