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Differential-Equations · Differential-Equations

Y' = 0.02*y*(100 - y)

\frac{d}{d x} y{\left(x \right)} = 0.02 \left(100 - y{\left(x \right)}\right) y{\left(x \right)}

Step by step

  1. \frac{d}{d x} y{\left(x \right)} = 0.02 \left(100 - y{\left(x \right)}\right) y{\left(x \right)}

    The differential equation.

  2. \text{order } 1

    Order 1: the highest derivative present.

  3. \text{Separable}

    Move everything in y to one side with dy and everything in x to the other, then integrate both sides.

  4. x + 0.5 \log{\left(1.0 y{\left(x \right)} - 100.0 \right)} - 0.5 \log{\left(1.0 y{\left(x \right)} \right)} = C_{1}

    General solution (C₁, C₂ … are arbitrary constants).

  5. \checkmark

    Verified: substituting the solution back into the equation gives 0.

Reveal the answer
x + 0.5 \log{\left(1.0 y{\left(x \right)} - 100.0 \right)} - 0.5 \log{\left(1.0 y{\left(x \right)} \right)} = C_{1}