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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.

مثال · y' + y = x

Differential-Equations · Differential-Equations

Y' + y = x

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

قدم ب قدم

  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.

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

اب آپ مسئلہ منتخب کريں يا ٹائپ کريں يا خود کريں ہر قدم ، تصوير ، جواب پوشيدہ هے جب تک آپ سوال نہ کريں

آپ کے اپنے کام رکھیں

مفت اکاؤنٹ تمام درسوں پر نوٹ شامل کر تا هے ، آپ نے جو ختم کر يا هے اس کا ريکورڈ ، آپ کے حل کيے گئے مسئلے ایک جگہ پر اور آپ اس صفحے کے بارے ميں سوال کر سکتے هيں تو تيرچر ريتھ ميں خود تمام لوگوں کے ليے کھلا هے ، خواہ آپ داخل هے يا نہيں

رجسٹر لاگ ان

علامتیں

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کیسے: 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.

سوالات لوگ پوچھتے ہیں

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.

میں زیادہ Differential Equations

Separable equationsSecond-order, constant coefficientsNonhomogeneous equationsModelling with differential equations