Linear Differential Equation

A Linear Differential Equation is a Differential Equation in which the dependent variable and its derivatives occur only in the first degree and no products of the dependent variable and its derivatives or of various order derivatives occur.

• Context:
  • It can range from being a Ordinary Linear Differential Equation to being a Partial Linear Differential Equation.
  • It can range from being a constant coefficient type to being a variable coefficient type.
  • It can be of the form [math]\displaystyle{ \frac{d^n y}{dx^n}+p_1(x) \frac{d^{n-1} y}{dx^{n-1}}+p_2(x)\frac{d^{n-2} y}{dx^{n-2}}+ \dots +p_{n-1}(x)\frac{dy}{dx}+p_n(x)=q(x) }[/math]
• Example(s):
  • [math]\displaystyle{ x^2 \frac{d^2 y}{dx^2}-x\frac{dy}{dx}+6y=log(x) }[/math] is a linear differential equation.
• Counter-Example(s):
  • [math]\displaystyle{ \frac{dy}{dx} \frac{d^2 y}{dx^2}+y^2=x^2 }[/math] is a Non-Linear Differential Equation.
• See: Ordinary Differential Equation, Partial Differential Equation, Vector Space.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/linear_differential_equation Retrieved:2015-1-31.
  • In mathematics, linear differential equations are differential equations having differential equation solutions which can be added together to form other solutions. They can be ordinary or partial. The solutions to linear equations form a vector space (unlike non-linear differential equations).
• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Ordinary_differential_equation#General_definition_of_an_ODE Retrieved:2015-1-31.
  • A differential equation is said to be linear if F can be written as a linear combination of the derivatives of y: :[math]\displaystyle{ y^{(n)} = \sum_{i=0}^{n-1} a_i(x) y^{(i)} + r(x) }[/math] where a_i(x) and r(x) are continuous functions in x. Non-linear equations cannot be written in this form. The function r(x) is called the source term, leading to two further important classifications:

    Homogeneous: If r(x) = 0, and consequently one "automatic" solution is the trivial solution, y = 0. The solution of a linear homogeneous equation is a complementary function, denoted here by y_c.

    Nonhomogeneous (or inhomogeneous): If r(x) ≠ 0. The additional solution to the complementary function is the particular integral, denoted here by y_p.

    The general solution to a linear equation can be written as y = y_c + y_p.

2011

• http://mathworld.wolfram.com/MatrixEquation.html
  • QUOTE: Nonhomogeneous matrix equations of the form :[math]\displaystyle{ \mathbf{A}\mathbf{x}=\mathbf{b} \tag{1} }[/math] can be solved by taking the matrix inverse to obtain :[math]\displaystyle{ \mathbf{x}=\mathbf{A}^{-1}\mathbf{b} \tag{2} }[/math] This equation will have a nontrivial solution iff the determinant [math]\displaystyle{ \operatorname{det}(\mathbf{A}) \neq 0 \tag{3} }[/math]. In general, more numerically stable techniques of solving the equation include Gaussian elimination, LU decomposition, or the square root method.

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