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ashok vardhan

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- Thread starter ashok vardhan
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In summary, the dimension of the solution set to a second order, homogeneous, linear differential equation is 2.

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ashok vardhan

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ashok vardhan

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Can anyone help me to clarify my doubt..

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HallsofIvy

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A differential equation does NOT have a "dimension". What you are asking about is the dimension of the

I assume you have seen the proof that the solution set does, in fact, form a vector space. You just need to observe if f and g are solutions, then, for any a and b,

(af+ bg)''+ p(x)(af+ bg)'+ q(x)(af+ bg)= af''+ bg''+ap(x)f'+ bp(x)g'+ aq(x)f+ bq(x)g= a(f''+ pf'+ qf)+ b(g''+ pg'+ qg)= a(0)+ b(0)= 0 so af+ bg is also a in that set.0

To see that "the set of all solutions to a second order, homogeneous, linear differential equation form a vector space of dimension 2", look at the inital value problems y''+ p(x)y'+ q(x)= 0, with y(a)= 1, y'(a)= 0 and with y(a)= 0, y'(a)= 1. Since y''= -p(x)y'- q(x)y, if p and q are continuous on some interval around x= a, then there exist a unique solution to each of those problems

Now let y(x) be any solution to the differential equation. Let A= y(a), B= y'(a). Then AY1+ BY2 also satisfies the differential equation and (AY1+ BY2)(a)= AY1(a)+ BY2(a)= A(1)+ B(0)= A and (AY1+ BY2)'(a)= AY1'(a)+ BY2'(a)= A(0)+ B(1)= B. Since AY1+ BY2 satisfies the same differential equation

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ashok vardhan

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sir, i don't know whether i can ask this question in this forum..if not please excuse me..my doubt is i want to learn Complex analysis in details from Basics..which is the best book??

- #5

blue_raver22

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The dimension of a 2nd order homogeneous equation is not always 2. It depends on the specific conditions and constraints of the equation. In general, the dimension can be determined by the number of independent variables and the order of the equation. In the case of a 2nd order homogeneous equation of the form y''+p(x)y'+q(x)=0, the dimension is 2 if p(x) and q(x) are continuous on a given interval I. However, if there are other constraints or discontinuities in the equation, the dimension may be different. It is important to carefully consider all factors when determining the dimension of an equation.

A 2nd order homogeneous equation is a mathematical equation that contains a second derivative of a variable and all terms in the equation are of the same degree. This means that the coefficients of all the terms must be constants and there should be no terms with a higher degree than 2. An example of a 2nd order homogeneous equation is y'' + 2y' + 3y = 0.

A 2nd order homogeneous equation is said to be homogeneous because all the terms in the equation have the same degree. This means that the equation is balanced and does not have any constant terms. In other words, the equation is symmetrical and can be solved without knowing the specific values of the variables involved.

The dimension of a 2nd order homogeneous equation depends on the number of independent variables present in the equation. For example, if the equation contains only one independent variable, the dimension would be 1. If the equation contains two independent variables, the dimension would be 2. The dimension can also be determined by the highest order derivative present in the equation.

No, a 2nd order homogeneous equation cannot have a dimension greater than 2. This is because the highest order derivative present in the equation is 2, so the dimension cannot be higher than that. However, the equation can have a dimension of 2 if it contains two independent variables or if the highest order derivative is multiplied by a constant.

The main difference between a 2nd order homogeneous equation and a 2nd order non-homogeneous equation is the presence of constant terms. A 2nd order homogeneous equation does not have any constant terms, whereas a 2nd order non-homogeneous equation can have constant terms. This means that the solutions to a homogeneous equation will always be in the form of exponential functions, while the solutions to a non-homogeneous equation can also include polynomial functions.

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