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Q9E

Expert-verifiedFound in: Page 332

Book edition
9th

Author(s)
R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages
616 pages

ISBN
9780321977069

**Find a general solution for the differential equation with x as the independent variable:**

$u\text{'}\text{'}\text{'}-9u\text{'}\text{'}+27u\text{'}-27u=0$

The general solution for the differential equation with x as the independent variable is $u\left(x\right)={c}_{1}{e}^{3x}+{c}_{2}x{e}^{3x}+{c}_{3}{x}^{2}{e}^{3x}$

In the equation, ${r}^{3}-9{r}^{3}+27r-27=0$ , we recognize a complete cube, namely, ${(r-3)}^{3}=0$ . Thus, it has just one root x = 3 of multiplicity three.

The general solution to the given differential equation is given by

$u\left(x\right)={c}_{1}{e}^{3x}+{c}_{2}x{e}^{3x}+{c}_{3}{x}^{2}{e}^{3x}$

Hence the final solution is $u\left(x\right)={c}_{1}{e}^{3x}+{c}_{2}x{e}^{3x}+{c}_{3}{x}^{2}{e}^{3x}$

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