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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.

ნამდვილი ასლი: y' + y = x

Differential-Equations · Differential-Equations

Solve y' + y = x

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

ჟრყოკა ოჲ ჟრყოკა.

  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.

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

Symbols used here

P(A),\ P(A \mid B)
probability, conditional probability
Chance of A; chance of A given that B happened.
f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\lim_{x \to a} f(x)
limit
The value f(x) approaches as x approaches a.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
y',\ y''
first and second derivative of y
Prime notation for derivatives with respect to x (or t).
C,\ C_1,\ C_2
arbitrary constants
Constants of integration fixed by initial conditions.
\frac{\partial u}{\partial t},\ \nabla^2 u
partial derivative in time, Laplacian
Rate of change in time; sum of second partials (the diffusion operator).

How to: 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.

Questions people ask

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.

ჲოთრაი ჟამ.

მეტი Differential Equations

Separable equationsSecond-order, constant coefficientsNonhomogeneous equationsModelling with differential equations