First-Order Linear Differential Equation

A First-Order Linear Differential Equation is a linear differential equation in which all the terms are linear and in which highest derivative (the order of the differential equation) is equal to 1.

• AKA: First-Order Ordinary Differential Equation.
• Context:
  • It is an equation of the form

[math]\displaystyle{ \frac{dy(t)}{dt}=f\left(t,y(t)\right)\qquad\iff\qquad\frac{dy}{dt}=f(t,y) }[/math]

with [math]\displaystyle{ f }[/math] representing all functions [math]\displaystyle{ y(t) }[/math] which satifies this equation. Conventionally, in differential equations [math]\displaystyle{ y(t) }[/math] is written simply as [math]\displaystyle{ y }[/math] as it's dependency in [math]\displaystyle{ t }[/math] is already implicit.

• It can be a expressed as

[math]\displaystyle{ a_1(t)\frac{dy}{dt}+a_0(t)y=b_0(t)\qquad\iff \qquad\frac{dy}{dt}+a(t)y=b(t) }[/math]

where [math]\displaystyle{ a_0, a_1, a=a_0/a_1 }[/math] are the coefficients of the differential equation and [math]\displaystyle{ b_0,b=b_0/a_1 }[/math] are linear functions. When [math]\displaystyle{ b_0, b=0 }[/math] the equation is called Homogeneous First-order Linear Differential Equation, otherwise it is called Inhomogeneous First-Order Linear Differential Equation. When coefficients [math]\displaystyle{ a_0, a_1 }[/math]are constants the equation is called First-Order Linear Differential Equation with Constant Coefficients

Alternatevelly, this expression can be written as

[math]\displaystyle{ a_1(t)y'+a_0(t)y=b_0(t)\quad\iff\quad y'+a(t)=b(t)\quad\textrm{with}\quad y'=\frac{dy}{dt} }[/math]

• It can range from being a Homogeneous First-order Linear Differential Equation with Constant Coefficients to being a Inhomogeneous First-Order Linear Differential Equation.

• Example(s):
  • Homogeneous First-Order Linear Differential Equation.
  • Inhomogeneous First-Order Linear Differential Equation.
• Counter-Example(s):
  • Second-order Linear Differential Equation.
  • Polynomial Equation.
• See: Linear Differential Equation, Linear Function, Differential Equation.

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References

2015

• (Wikipedia, 2015) ⇒

1999

• (Wolfram Mathworld , 1999) ⇒

1992

• (Martin Braun, 1992)) ⇒ Martin Braun (1974, 1977 1982, 1992) "Differential Equations and their Applications", Spring-Verlag New York, Inc. ⇒ http://www.springer.com/us/book/
  • QUOTE: See Chapter 1

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