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.

דוגמה עובדת · y' + y = x

Differential-Equations · Differential-Equations

Y' + y = x

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

צעד אחר צעד

  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.

גלה את התשובה
y{\left(x \right)} = C_{1} e^{- x} + x - 1

עכשיו אתה בחר בעיה, או הקלד או צייר בעצמך. כל צעד, תמונה, התשובה חבויה עד שתשאל.

תמשיך לעבוד.

חשבון חינם מוסיף הערות על כל שיעור, תיעוד של מה שסיימת, הבעיות שלך נפתרות במקום אחד, ומורה פרטי אתה יכול לשאול על הדף הזה. המתמטיקה עצמה פתוחה לכולם, חתומה או לא.

הרשמה התחברות

סמלים המשמשים כאן

הקש כל סמל להגדרה המלאה, תמונה, ומה כל אות בזה אומר.

איך: 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.

שאלות שאנשים שואלים

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.

יותר בפנים. Differential Equations

Separable equationsSecond-order, constant coefficientsNonhomogeneous equationsModelling with differential equations