JosephK said:Is a linear equation y'+P(x)y=Q(x) not linear if P(x) and Q(x) are not linearly dependent function?
Does linearly dependent mean a constant multiplied by P(x) will equal Q(x)?
Thank you.
Correct.JosephK said:In recognizing linear differential equations for example y'+3x^2y=x^2 I do not say this linear differential equation is linear because I can multiply x^2 by 3).
I should say because y and y' are of the first degree this equation is linear.
Correct.What about y'-y=xy^2?
Is this linear differential equation non-linear because the y to the right is not in the first degree?
It's non-linear because a non-linear function of y (y * ln y) appears in the equation.What about y'-2xy=1/x*y*lny?
Is this linear differential equation not linear because the coefficient of the y to the right hand side depends on y?
Yes. This is a linear differential equation. A linear DE is one in which y, y', y'', etc. occur to the first power. How the independent variable (x in this case) occurs doesn't enter into the description.JosephK said:In recognizing linear differential equations for example y'+3x^2y=x^2 I do not say this linear differential equation is linear because I can multiply x^2 by 3).
I should say because y and y' are of the first degree this equation is linear.
No. This differential equation is nonlinear for the reason you say. A differential equation is either linear or nonlinear. It makes no sense to ask if a linear DE is nonlinear. If it's nonlinear it is not a linear differential equation.JosephK said:What about y'-y=xy^2?
Is this linear differential equation non-linear because the y to the right is not in the first degree?
Your question should be, "Is this differential equation nonlinear ..." It is nonlinear because of the lny factor on the right side.JosephK said:What about y'-2xy=1/x*y*lny?
Is this linear differential equation not linear because the coefficient of the y to the right hand side depends on y?
Linear independence refers to a set of vectors in a vector space that cannot be expressed as a linear combination of each other. In other words, the vectors are not dependent on each other and contribute unique information to the space.
A set of vectors is linearly independent if and only if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0, where c1, c2, ..., cn are constants and v1, v2, ..., vn are the vectors in the set.
A set of vectors is linearly independent if and only if it spans the entire vector space, meaning that any vector in the space can be expressed as a linear combination of the vectors in the set. If a set of vectors is not linearly independent, it can only span a subspace of the vector space.
Yes, a set of two vectors can be linearly dependent if one vector is a scalar multiple of the other. For example, if v1 = [1, 2] and v2 = [2, 4], these vectors are linearly dependent because v2 is equal to 2v1.
In solving systems of linear equations, linear independence is used to determine the number of unique solutions. If the system has a unique solution, the set of equations is linearly independent. If the system has infinitely many solutions, the equations are linearly dependent. This concept is also used in finding the basis of a vector space, which is important in solving systems of linear equations.