Second-Order Linear Differential Equation

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A Second-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 2.

[math]\displaystyle{ \frac{d^2y(t)}{dt^2}=f\left(t,y(t)\frac{dy(t)}{dt}\right)\qquad\iff\qquad\frac{d^2y}{dt^2}=f\left(t,\frac{dy}{dt}\right) }[/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_2(t)\frac{d^2y}{dt^2}+a_1(t)\frac{dy}{dt}+a_0(t)y=b(t)\qquad\textrm{or}\qquad\frac{d^2y}{dt^2}+p(t)\frac{dy}{dt}+q(t)y=g(t) }[/math]
where [math]\displaystyle{ a_0, a_1, a_2, p=a_1/a_2,q=a_0/a_2 }[/math] are the coefficients of the differential equation and [math]\displaystyle{ b,g=b/a_2 }[/math] are linear functions. When [math]\displaystyle{ b,g=0 }[/math] the equation is called Homogeneous Second-Order Linear Differential Equation, otherwise it is called Inhomogeneous Second-Order Linear Differential Equation. When coefficients [math]\displaystyle{ a_0, a_1, a_2 }[/math]are constants the equation is called Second-Order Linear Differential Equation with Constant Coefficients
Alternatevelly, this expression can be written as
[math]\displaystyle{ a_2(t)y''+a_1(t)y'+a_0(t)y=b(t)\quad\textrm{or} \quad y''+p(t)y'+q(t)y=g(t)\quad\textrm{with}\quad y''=\frac{d^2y}{dt^2},\;y'=\frac{d^2y}{dt} }[/math].

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