Question

classify the following DE's:
1-\[dy+xydx=0\]
2-\[(1+x^2)dx+2x \sin (x)dy=x^2\]
3-\[(\frac{ d^3y }{ dx^3 })^2+(\frac{ d^2y }{ dx^2})+y^7=y\]
4-\[yy'''+xy''+x^2y'=y\]
5-\[y'''+xy''+x^2y'=e^y\]
6-\[y'''+xy''+x^2y'=y\]
7-\[\frac{ d^2y }{ dx^2 }+y=0\]
8-(\[(\frac{ d^2y }{ dx^2 })^5+y=0\])
9-\[(\frac{ d^3y }{ dx^3})^2+(\frac{ d^2y }{ dx^2 })^5+y^7=y\]
10-\[y^2(y'''+y''+xy')=\sin ^2(x)\]

Answer 1

i take it you're just looking for something like (A) linear or not and (B) homogeneous or not? maybe the order? that right

anyways, you should probably put down what you think first for each .......

Answer 2

ok all of these equations are ordinary differential equations right?

Answer 3

yes - ordinary, no partial derivatives in there as far as i can see

Answer 4

1) order:1-Degree:1
2) order:1-Degree:1
3) order:3-Degree:2
4) order:3-Degree:1
5) order:3-Degree:1
6) order:3-Degree:1
7) order:2-Degree:1
8) order:2-Degree:5
9) order:3-Degree:2
10) order:3-Degree:1
is this right ? and now i need the linearity and homogeneity?

Answer 5

re- order, you take that from the order of the derivative so y' is first order, y'' is second and so on

you've put smileys in which is why i have said this

so, to illustrate, for 5) & 9), to take but two, you have 3rd order....

make sense?!

PS i think 7) and 8) need to be re-examined in this light.

Answer 6

The degree of a differential equation is the power of the highest order derivative so i would sort out the order of the smileys and the others, before moving to degree

Answer 7

okay , all my answers are right what about the linearity and homogeneity?

Answer 8

A linear ODE is one that can be written in the form
\[f_n(x)\frac{d^ny}{dx^n}+\cdots+f_1(x)\frac{dy}{dx}+f_0(x)y=g(x)\]
Notice that all the derivatives of \(y\) have a power of no more/less than \(1\) \(\bigg(\)i.e. you don't see any terms like \(\left.\left(\dfrac{dy}{dx}\right)^2\right)\). Analogously, an algebraic equation is linear in a variables \(t_1,\ldots,t_n\) if you can write an equation in the form
\[\quad\quad\quad\quad a_1t_1+\cdots+a_nt_n=c\]where \(a_1,\ldots,a_n,c\) are constants.

Also notice that there are no functions of the dependent variable \(y\) and its derivatives, such as \(\dfrac{1}{y}\) or \(\cos \left(y\dfrac{dy}{dx}\right)\), etc.

Answer 9

As for homogeneity, you have to be a bit more specific, as there are two different meanings.
https://en.wikipedia.org/wiki/Homogeneous_differential_equation