Linear DEs

DEs where the dependent variable appears "linearly"

• $y,y^{\prime },y^{″}\dots$ never appear multiplied/divided together or raised to any power other than 1
• terms including y and its derivatives may only be multiplied by constants or functions of the independent variable and added/subtracted from one another
• equations including functions of y or its derivatives are not linear ($\sin (y),e^y$)

$\frac{d^{3}x}{d{t}^3}-x\frac{dx}{dt}=\csc (t+\pi )$

• nonlinear, x is being multiplied by x'

$\frac{1}{x}{u}^{(4)}-u^{\prime }=-x$

• linear

Standard form for linear DEs