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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.

İşlənmiş nümunə · y' + y = x

Differential-Equations · Differential-Equations

Y' + y = x

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

Addım-addım

  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.

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

İndi siz Problem seçin, ya da özünü yazın ya da çəkin. Hər addım, bir rəsm, cavab siz soruşana qədər gizli qalır.

Özün işlə

Pulsuz hesab hər dərsdə qeydlər əlavə edir, bitirdiyiniz işlərin qeydini saxlayır, həll etdiyiniz problemləri bir yerdə saxlayır və bu səhifə haqqında soruşa biləcəyiniz müəllimi də əlavə edir. Riyaziyyat özü daxil olub-olmadığınıza baxmayaraq hər kəsə açıqdır.

Qeyd Et Giriş

Burada istifadə edilən simvollar

Tam təsviri, şəkli və hər hərfin nə demək olduğunu görmək üçün hər hansı simvolu vurun.

Növbəti: First-order linear equations

  1. The differential equation.
  2. Order 1: the highest derivative present.
  3. 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. Characteristic equation of the homogeneous part (substitute y = e^{rx}).
  5. Its roots.
  6. General solution (C₁, C₂ … are arbitrary constants).
  7. Verified: substituting the solution back into the equation gives 0.

İnsanlar soruşurlar

What is a differential equation?

An equation whose unknown is a function, relating it to its own derivatives. "The rate of growth is proportional to the population" is y′ = ky, and solving it means finding y as a function of time.

Why does the solution have arbitrary constants?

Integrating loses information: many functions share the same derivative. An n-th order equation has n constants, fixed by n initial or boundary conditions.

Daha çox Differential Equations

Separable equationsSecond-order, constant coefficientsNonhomogeneous equationsModelling with differential equations