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Differential-Equations · Differential-Equations

Y' = -4y

\frac{d}{d x} y{\left(x \right)} = - 4 y{\left(x \right)}

Step by step

  1. \frac{d}{d x} y{\left(x \right)} = - 4 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. r + 4 = 0

    Characteristic equation of the homogeneous part (substitute y = e^{rx}).

  5. r = -4

    Its roots.

  6. y{\left(x \right)} = C_{1} e^{- 4 x}

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

  7. \checkmark

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

Reveal the answer
y{\left(x \right)} = C_{1} e^{- 4 x}