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First-order linear equations

y′ + P(x)y = Q(x) and the integrating factor that makes it integrable.

Multiply y′ + P(x)y = Q(x) by μ = e^{∫P dx} and the left side collapses to (μy)′ — one integration finishes it. This is the workhorse for mixing tanks, RC circuits and Newton's law of cooling.

ਕੰਮ ਉਦਾਹਰਨ: y' + y = x

Differential-Equations · Differential-Equations

Solve y' + y = x

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

ਕਦਮ ਦਰ ਕਦਮ

  1. y{\left(x \right)} + \frac{d}{d x} y{\left(x \right)} = x

    The differential equation.

  2. \text{order } 1

    Order 1: the highest derivative present.

  3. \text{Exact}

    M dx + N dy = 0 with ∂M/∂y = ∂N/∂x: find the potential F with F_x = M, F_y = N; the solution is F = C.

  4. r + 1 = 0

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

  5. r = -1

    Its roots.

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

    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^{- x} + x - 1

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ਹੋਰ ਵਿੱਚ Differential Equations

Separable equationsSecond-order, constant coefficientsNonhomogeneous equationsModelling with differential equations