Definition:Linear First Order Ordinary Differential Equation/Constant Coefficients
< Definition:Linear First Order Ordinary Differential Equation(Redirected from Definition:Linear First Order ODE with Constant Coefficients)
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Definition
A linear first order ordinary differential equation with constant coefficients is a linear first order ordinary differential equation which is in (or can be manipulated into) the form:
- $\dfrac {\d y} {\d x} + a y = \map Q x$
where:
It is:
- Linear because both $\dfrac {\d y} {\d x}$ and $y$ appear to the first power, and do not occur multiplied together
- First order because the highest derivative is $\dfrac {\d y} {\d x}$
- Ordinary because there are no partial derivatives occurring in it.
Also see
- Solution to Linear First Order ODE with Constant Coefficients: Its general solution is:
- $\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x + C}$
- where $C$ is an arbitrary constant.
- Results about linear first order ODEs with constant coefficients can be found here.
Sources
- 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 1$. The first order equation: $\S 1.1$ Introduction: $(1)$