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Found in: Page 332

Fundamentals Of Differential Equations And Boundary Value Problems

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

See the step by step solution

Step 1: Auxiliary equation:

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

Step 2: General solution:

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