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Separable equations

dy/dx = f(x)g(y): separate, integrate both sides, solve for y.

If the rate of change factors into a piece in x times a piece in y, you can move all the y's to one side with dy and integrate both sides. y′ = 2y gives y = Ce^{2x} — exponential growth, the shape of populations, compound interest and radioactive decay (with a negative constant). The plot shows the family of solutions for several C.

Esempio di funzionamento: y' = 2y

Differential-Equations · Differential-Equations

Solve y' = 2y

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

Passo dopo passo

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

    The differential equation.

  2. \text{order } 1

    Order 1: the highest derivative present.

  3. \text{Separable}

    Move everything in y to one side with dy and everything in x to the other, then integrate both sides.

  4. r - 2 = 0

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

  5. r = 2

    Its roots.

  6. y{\left(x \right)} = C_{1} e^{2 x}

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

  7. \checkmark

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

Rivela la risposta
y{\left(x \right)} = C_{1} e^{2 x}

Symbols used here

f'(x),\ \frac{dy}{dx}
derivative
Instantaneous rate of change; slope of the graph.
y',\ y''
first and second derivative of y
Prime notation for derivatives with respect to x (or t).
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).
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: Separable equations

  1. The differential equation.
  2. Order 1: the highest derivative present.
  3. Move everything in y to one side with dy and everything in x to the other, then integrate both sides.
  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.

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Più in Differential Equations

First-order linear equationsSecond-order, constant coefficientsNonhomogeneous equationsModelling with differential equations