maths.free › Differential Equations › First-order linear equations

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.

Contoh yang dikerjakan · y' + y = x

Differential-Equations · Differential-Equations

Y' + y = x

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

Langkah demi langkah

  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.

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

Sekarang kau Pilih masalah, atau taip atau lukis sendiri. Setiap langkah, gambar, jawapan tersembunyi sehingga anda tanya.

Jaga kerja awak sendiri.

Akaun percuma menambah nota pada setiap pengajaran, rekod apa yang anda selesaikan, masalah yang anda selesaikan di satu tempat, dan seorang guru yang anda boleh tanya tentang halaman ini. Matematik itu sendiri terbuka kepada semua orang, sama ada anda log masuk atau tidak.

Daftar Masuk Log masuk

Simbol yang digunakan di sini

Tekan simbol mana-mana untuk definisi penuh, gambar, dan apa makna setiap huruf dalamnya.

Bagaimana untuk: 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.

Soalan yang orang tanya

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.

Lebih dalam Differential Equations

Separable equationsSecond-order, constant coefficientsNonhomogeneous equationsModelling with differential equations