# Are solultions of D.E. a Vector Space?

#### coldturkey

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Let V be the solutions to the differential equation:

$a_{1}y' + a_{0} = x^2 + e^x$

Decide using the properties of pointwise addition and scalar multiplication if V is a vector space or not.

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Ok im having real trouble with this question. I can prove is something is a vector space aslong as I know what im trying to proove.

To solve this question do I need to find or make up y? Or am I just assuming V = (a0, a1) and need to prove something like
$$(a1 + b1)y' + (a0+b0) = x^2 + e^x + b1y' + b0$$

#### HallsofIvy

The fundamental property of a vector space is "linearity": if u and v are vectors, then so is au+ bv for any numbers a and b.

Suppose y1 and y2 are solutions to that differential equation (that is they satisfy the equation) can you show that, for any number a and b, ay1+ by2 also satisfies the differential equation?

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