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Separable equations

dy/dx = f(x)g(y): separate, integrate both sides, solve for y.

If the rate of change factors into a piece in x times a piece in y, you can move all the y's to one side with dy and integrate both sides. y′ = 2y gives y = Ce^{2x} — exponential growth, the shape of populations, compound interest and radioactive decay (with a negative constant). The plot shows the family of solutions for several C.

ਕੰਮ ਉਦਾਹਰਨ: y' = 2y

Differential-Equations · Differential-Equations

Solve y' = 2y

\frac{d}{d x} y{\left(x \right)} = 2 y{\left(x \right)}

ਕਦਮ ਦਰ ਕਦਮ

  1. \frac{d}{d x} y{\left(x \right)} = 2 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 - 2 = 0

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

  5. r = 2

    Its roots.

  6. y{\left(x \right)} = C_{1} e^{2 x}

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

  7. \checkmark

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

ਜਵਾਬ ਦਿਓ
y{\left(x \right)} = C_{1} e^{2 x}

ਆਪਣਾ ਹੀ ਕੋਸ਼ਿਸ਼ ਕਰੋ

ਹੋਰ ਵਿੱਚ Differential Equations

First-order linear equationsSecond-order, constant coefficientsNonhomogeneous equationsModelling with differential equations