Homogeneous definition for first order differential equation and higher order differential equations?
$begingroup$
For first order https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_first-order_differential_equations
and for higher order, it is https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_linear_differential_equations
So, using the definition for higher order can we prove a first order differential equation is homogeneous?
For an example;
$ 4frac{dy}{dx}+y=0 $ is homogeneous considering definition for higher order linear D.E. But using the original definition for homogeneity of first order we get
$frac{dy}{dx}=frac{-y}{4}$ which does not seem to be homogeneous.
I saw definition for higher order being used for first order as well, so got confused. Here is a snap of the video, where N=non homogeneous and H=homogeneous.
calculus ordinary-differential-equations homogeneous-equation
asked Dec 12 '18 at 13:02
Abbas MiyaAbbas Miya
16011
$endgroup$
add a comment |
$begingroup$
For first order https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_first-order_differential_equations
and for higher order, it is https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_linear_differential_equations
So, using the definition for higher order can we prove a first order differential equation is homogeneous?
For an example;
$ 4frac{dy}{dx}+y=0 $ is homogeneous considering definition for higher order linear D.E. But using the original definition for homogeneity of first order we get
$frac{dy}{dx}=frac{-y}{4}$ which does not seem to be homogeneous.
I saw definition for higher order being used for first order as well, so got confused. Here is a snap of the video, where N=non homogeneous and H=homogeneous.
calculus ordinary-differential-equations homogeneous-equation
asked Dec 12 '18 at 13:02
Abbas MiyaAbbas Miya
16011
$endgroup$
add a comment |
$begingroup$
For first order https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_first-order_differential_equations
and for higher order, it is https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_linear_differential_equations
So, using the definition for higher order can we prove a first order differential equation is homogeneous?
For an example;
$ 4frac{dy}{dx}+y=0 $ is homogeneous considering definition for higher order linear D.E. But using the original definition for homogeneity of first order we get
$frac{dy}{dx}=frac{-y}{4}$ which does not seem to be homogeneous.
I saw definition for higher order being used for first order as well, so got confused. Here is a snap of the video, where N=non homogeneous and H=homogeneous.
calculus ordinary-differential-equations homogeneous-equation
asked Dec 12 '18 at 13:02
Abbas MiyaAbbas Miya
16011
$endgroup$
For first order https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_first-order_differential_equations
and for higher order, it is https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_linear_differential_equations
So, using the definition for higher order can we prove a first order differential equation is homogeneous?
For an example;
$ 4frac{dy}{dx}+y=0 $ is homogeneous considering definition for higher order linear D.E. But using the original definition for homogeneity of first order we get
$frac{dy}{dx}=frac{-y}{4}$ which does not seem to be homogeneous.
I saw definition for higher order being used for first order as well, so got confused. Here is a snap of the video, where N=non homogeneous and H=homogeneous.
calculus ordinary-differential-equations homogeneous-equation
calculus ordinary-differential-equations homogeneous-equation
asked Dec 12 '18 at 13:02
Abbas MiyaAbbas Miya
16011
asked Dec 12 '18 at 13:02
Abbas MiyaAbbas Miya
16011
edited Dec 12 '18 at 13:07
Abbas Miya
asked Dec 12 '18 at 13:02
Abbas MiyaAbbas Miya
16011
asked Dec 12 '18 at 13:02
Abbas MiyaAbbas Miya
16011
asked Dec 12 '18 at 13:02
Abbas MiyaAbbas Miya
16011
16011
add a comment |
add a comment |
1 Answer
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An equation is homogeneous if it is linear in $x$ and its derivatives. Linear here means not affine, but in the sense that $x=0$ is an element of the solution space.
It does not matter what equivalent form of the equation you use (as long as it stays linear).
answered Dec 12 '18 at 14:03
LutzLLutzL
58.5k42054
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
An equation is homogeneous if it is linear in $x$ and its derivatives. Linear here means not affine, but in the sense that $x=0$ is an element of the solution space.
It does not matter what equivalent form of the equation you use (as long as it stays linear).
answered Dec 12 '18 at 14:03
LutzLLutzL
58.5k42054
$endgroup$
add a comment |
$begingroup$
An equation is homogeneous if it is linear in $x$ and its derivatives. Linear here means not affine, but in the sense that $x=0$ is an element of the solution space.
It does not matter what equivalent form of the equation you use (as long as it stays linear).
answered Dec 12 '18 at 14:03
LutzLLutzL
58.5k42054
$endgroup$
add a comment |
$begingroup$
An equation is homogeneous if it is linear in $x$ and its derivatives. Linear here means not affine, but in the sense that $x=0$ is an element of the solution space.
It does not matter what equivalent form of the equation you use (as long as it stays linear).
answered Dec 12 '18 at 14:03
LutzLLutzL
58.5k42054
$endgroup$
An equation is homogeneous if it is linear in $x$ and its derivatives. Linear here means not affine, but in the sense that $x=0$ is an element of the solution space.
It does not matter what equivalent form of the equation you use (as long as it stays linear).
answered Dec 12 '18 at 14:03
LutzLLutzL
58.5k42054
answered Dec 12 '18 at 14:03
LutzLLutzL
58.5k42054
answered Dec 12 '18 at 14:03
LutzLLutzL
58.5k42054
answered Dec 12 '18 at 14:03
LutzLLutzL
58.5k42054
58.5k42054
add a comment |
add a comment |
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