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On Pauls notes webpage, there is the following problem to be solved by variation of parameters:

[itex]ty''-(t+1)y'+y=t^2[/itex] (1)

On the page, the fundamental set of solutions if formed on the basis of the complementary solution. The set is:

[itex]y_{1}(t)=e^t[/itex] and [itex]y_{2}(t)=t+1[/itex]

Now, I must be missing something here. Since I get the complementary solution for the homogeneous equation of (1):

[itex]r=\frac{(t+1)+/- \sqrt{(t+1)^2-4t}}{2t}[/itex] which solves as [itex]r_{1}=1[/itex] and [itex]r_{2}=\frac{1}{t}[/itex] which would give a complementary solution of:

[itex]Y_{c}=C_{1}e^{t}+C_{2}e^{\frac{1}{t}t}=C_{1}e^{t}+C_{2}e^1[/itex]

from which I would get [itex]y_{1}(t)=e^t[/itex] and [itex]y_{2}(t)=e[/itex]

What have I missed, must be simple...

Regards,

U.

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# Problem with finding the complementary solution of ODE

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