What does it mean for a differential equation to be autonomous?

1 answer:

7 0

The term "autonomous" refers to an ordinary differential equation that relates the derivatives of the dependent variable as a function *only* of the dependent variable. In other words, the ODE doesn't explicitly depend on the independent variable.

Examples:

$y'=y^2-1$ is autonomous

$y'=y^2-1+x$ is *not* autonomous

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WILL GIVE BRAINLIEST! NEED QUICK HELP ASAP! Photo added! least common denominator x^3/x^2+4x+4 and -9/x^2+5x+6

Oksanka [162]

The denominator( s ) we are given are $x^2 + 4x + 4$ , and . The first thing we want to do is factor the expressions, to make this easier -

$x^2 + 4x + 4$

This expression is a perfect square, as ( x )^2 = x^2, ( 2 )^2 = 4, 2 * ( x ) * ( 2 ) = 4x. Thus, the simplified expression should be the following -

$( x + 2 )^2$

The other expression is, on the other hand, not a perfect square so we must break this expression into groups and attempt factorization -

$x^2 + 5x + 6,\\\left(x^2+2x\right)+\left(3x+6\right),\\x\left(x+2\right)+3\left(x+2\right),\\\left(x+2\right)\left(x+3\right)\\\\Factored Expression = \left(x+2\right)\left(x+3\right)$