Get constants from DE and special solution

[lethal quail]

$y''+\alpha y'+\beta y=\gamma e^x$ \
Special solution:
$y^*=e^{2x}+(1+x)e^x$ \
Obtain constants $\alpha$, $\beta$ and $\gamma$

[vale relic]

FishBone.EarthKhan

[lethal quail]

I know how to do it, by substiting y* into the DE, by the answer says there's a faster way

The answer says just by looking at the special solution you can know the roots are r_1=1 and r_2=2

But ehhh....how did that work

I mean, shouldn't the special solution look like:

$y^*=x^kR_m(x)e^{\lambda x}$ \
where $\lambda = 1$?

[vale relic]

FishBone.EarthKhan

[lethal quail]

Where did that e^{2x} come from

[lethal quail]

I got it

the general solution is

$y=C_1e^x+C_2e^{2x}$

[vale relic]

FishBone.EarthKhan

[lethal quail]

That's why you know the roots after just looking at it

.solved