Inhomogeneous First-Order Linear Differential Equation

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

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

[math]\displaystyle{ a_1(t)\frac{dy}{dt}+a_0(t)y=b(t)\quad\iff \quad\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_1(t)y'+a_0(t)y=b(t)\quad\iff \quad y'+q(t)y=g(t) }[/math]

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

• Example(s):
  • Radioactive decay with a number of disintegrations
• Counter-Example(s):
  • Homogeneous Second-Order Linear Differential Equation.
  • Inhomogeneous Second-Order Linear Differential Equation.
  • Homogeneous 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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