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Second-order, constant coefficients

ay″ + by′ + cy = 0 and the characteristic equation: oscillation, damping, growth.

Try y = e^{rx} and the equation becomes a quadratic in r — the characteristic equation. Real distinct roots give exponentials, a repeated root adds an x, complex roots give sines and cosines. y″ + y = 0 is the simple harmonic oscillator: every pendulum, spring and LC circuit.

Exemplu lucrat: y'' + y = 0

Differential-Equations · Differential-Equations

Solve y'' + y = 0

y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0

Pas cu pas

  1. y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0

    The differential equation.

  2. \text{order } 2

    Order 2: the highest derivative present.

  3. \text{Constant coefficients, homogeneous}

    Try y = e^{rx}: the characteristic polynomial in r gives the roots, and each root contributes a term to the general solution.

  4. r^{2} + 1 = 0

    Characteristic equation of the homogeneous part (substitute y = e^{rx}).

  5. r = - i,\ r = i

    Its roots.

  6. y{\left(x \right)} = C_{1} \sin{\left(x \right)} + C_{2} \cos{\left(x \right)}

    General solution (C₁, C₂ … are arbitrary constants).

  7. \checkmark

    Verified: substituting the solution back into the equation gives 0.

Revelează răspunsul
y{\left(x \right)} = C_{1} \sin{\left(x \right)} + C_{2} \cos{\left(x \right)}

Symbols used here

\sin,\ \cos,\ \tan
sine, cosine, tangent
Ratios of sides in a right triangle; coordinates on the unit circle.
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.
i
imaginary unit
i² = −1.
\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: Second-order, constant coefficients

  1. The differential equation.
  2. Order 2: the highest derivative present.
  3. Try y = e^{rx}: the characteristic polynomial in r gives the roots, and each root contributes a term to the general solution.
  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.

Încearcă pe tine.

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