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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: ${y}{\text{'}}{\text{'}}{\text{'}}{-}{y}{\text{'}}{\text{'}}{+}{2}{y}{=}{0}$

The general solution for the differential equation with x as the independent variableis .$y\left(x\right)={c}_{1}{e}^{-x}+{c}_{2}{e}^{-5x}+{c}_{3}{e}^{4x}$

See the step by step solution

Step 1: Auxiliary equation:

The given differential equationis $y\text{'}\text{'}\text{'}-y\text{'}\text{'}+2y=0$. To solve this equation, we look at its auxiliary equation which is ${m}^{3}-{m}^{2}+2=0$ . Observe that -1 is a solution of this equation. So,

${m}^{3}-{m}^{2}+2=\left(m+1\right)\left({m}^{2}-2m+2\right)$

Step 2: Inspecting the sum further:

To get the other two roots of auxillary equation, we need to solve ${m}^{3}-{m}^{2}+2=0$ . We have,

$m=\frac{2±\sqrt{4-8}}{2}=1±i$

Step 3: General solution:

We have m = -1,$1±i$ .From (7) of 328 and (18) of page 330, we conclude that the general solution of the given differential equation is $y={C}_{1}{e}^{-x}+{C}_{2}{e}^{x}\mathrm{cos}x+{C}_{3}{e}^{x}\mathrm{sin}x$ where ${C}_{1},{C}_{2},{C}_{3}$ are arbitrary constants.

The solution of the given differential equation is$y={C}_{1}{e}^{-x}+{C}_{2}{e}^{x}\mathrm{cos}x+{C}_{3}{e}^{x}\mathrm{sin}x$ , where ${C}_{1},{C}_{2},{C}_{3}$ are arbitrary constant.

Hence the final solution is $y\left(x\right)={c}_{1}{e}^{-x}+{c}_{2}{e}^{-5x}+{c}_{3}{e}^{4x}$

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