User:Oh Isaac/R6-9

Given:
1.L2-ODE-VC:

2.Trial solution:

3.Characteristic equation:

the roots are:

Find:

 * 1) select a valid homogeneous solution, and call it $$u_1$$.
 * 2) Find the 2nd homogeneous solution $$u_2(x)$$ by variation of parameters, and compare to $$e^{xr_2(x)}$$.

Solution

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On our honor, we did this assignment on our own, without looking at the solutions in previous semesters or other online solutions.
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1.select 1st homogeneous solution: The valid root is $$r_1=1$$, so the valid 1st homogenous solution is $$e^{xr_1}=e^x$$.

2.Find 2nd homogeneous solution by variation of parameters. We get the operation of variation of parameters as follows.

Now substitute the terms in $$with that from $$:

follow the operations from$$to $$, then we get:

So the 2nd homogeneous solution is $$u_2(x)=x$$.

Comparing to $$e^{xr_2}=e^{\frac{x}{x-1}}$$ reflects that $$r_2(x)$$ is not a valid root.