Aryan Emery

2022-02-24

Define what linear first-order equation forms?

Mikayla Swan

Linear differential equation:
a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
${a}_{0}\left(x\right)y+{a}_{1}\left(x\right){y}^{\prime }+{a}_{2}\left(x\right)y{}^{″}+\dots +{a}_{n}\left(x\right){y}^{\left(n\right)}+b\left(x\right)=0$,
where ${a}_{0}\left(x\right),\dots {a}_{n}\left(x\right)$ and $b\left(x\right)$ are arbitrary differentiable functions that do not need to be linear, and ${y}_{1}^{\prime },\dots ,{y}^{\left(n\right)}$ are the successive derivatives of an unknown function y of the variable x.
This is an ordinary differentiable function (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives.
linear first-order equation forms
$\frac{dy}{dx}+Py=Q$
Where P and Q are functions of x.
This is the required form.

Donald Erickson