Inhomogeneous Second-Order Linear Differential Equation

A Inhomogeneous Second-Order Linear Differential Equation is a Second-order Linear Differential Equation which equals to a non-zero function.

• AKA: Nonhomogeneous Second-Order Linear Differential Equation.
• Context:
  • It can be expressed as

[math]\displaystyle{ a_2(t)\frac{d^2y}{dt^2}+a_1(t)\frac{dy}{dt}+a_0(t)y=b(t)\quad\iff \quad\frac{d^2y}{dt^2}+p(t)\frac{dy}{dt}+q(t)y=g(t) }[/math]

[math]\displaystyle{ \textrm{with} \quad b(t)\neq 0, \quad\textrm{and}\quad g(t)\neq 0 }[/math]

Alternatively, these expressions can be written as

[math]\displaystyle{ a_2(t)y''+a_1(t)y'+a_0(t)y=b(t)\quad\iff \quad y''+p(t)y'+q(t)y=g(t) }[/math]

[math]\displaystyle{ \textrm{with}\quad y''=\frac{d^2y}{dt^2},\;y'=\frac{dy}{dt} \quad\textrm{and}\quad b(t)\neq 0, \quad g(t)\neq 0 }[/math]

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

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