Independent Solutions of a Differential Equations?

Question

Determine the differential equation whose set of independent solution is $\{e^x,xe^x,x^2e^x\}$? What is the meaning of independent solutions here?

Answer 1

They mean that the general solution of the equation is

$$y=C_1e^x+C_2xe^x+C_3x^2e^x.$$

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When you have terms that are the product of increasing powers of $x$ times the same exponential, this corresponds to multiple roots of the characteristic polynomial.

In this case, a triple root $1$ corresponds to

$$\left(\frac d{dx}-1\right)^3y=y'''-3y''+3y-y.$$

As you can check by direct subtitution, all above terms are solutions.

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We can explain this result as follows. Consider the operator $D:=\dfrac d{dx}-1$, such that

$$Dy=y'-y.$$

As

$$(ye^{-x})'=(y'-y)e^{-x}$$ we can write

$$Dy=(ye^{-x})'e^x.$$

Then iterating,

$$D^2y=((ye^{-x})'e^xe^{-x})'e^x=(ye^{-x})''e^x$$ and

$$D^3y=((ye^{-x})''e^xe^{-x})'e^x=(ye^{-x})'''e^x.$$

Now the solution of

$$D^3y=0$$ is that of

$$(ye^{-x})'''=0$$ or

$$y=(C_0+C_1x+C_2x^2)e^x.$$