Let's say we have a homogeneous differential equation y'(x) + p(x)y = 0 on an interval [a,b]. Hw do we check whether its set of solutions forms a vector space or not? And if it forms a vector space then what will be its dimension?
What is the dimension of solution set of a homogeneous linear equation?
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linear-algebra
vector-spaces
1 Answers
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To show that the solutions of the differential equation form a linear space we have to check that it is closed under addition and scalar multiplication. So if y1 is a solution then cy1( where c is a constant) is also a solution of the differential equation. You can also see that if y1 and y2 are solutions of the differential equations then their sum is also a solution of the equation. Hence the solution space of a linear homogeneous equation is a vector space.