Are solultions of D.E. a Vector Space?

[coldturkey]

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Let V be the solutions to the differential equation:

<br /> a_{1}y&#039; + a_{0} = x^2 + e^x<br />

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

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Ok I am having real trouble with this question. I can prove is something is a vector space aslong as I know what I am 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
<br /> (a1 + b1)y&#039; + (a0+b0) = x^2 + e^x + b1y&#039; + b0<br />

Could someone please help me get started? Thanks

 

[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?