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### 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

# Derive the system (7) in the special case when ${\mathbit{n}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{\mathbf{3}}$. [Hint: To determine the last equation, require that ${\mathbit{I}}{\mathbf{.}}\left[{y}_{p}\right]{\mathbf{=}}{\mathbit{g}}$ and use the fact that ${{\mathbit{y}}}_{{\mathbf{1}}}{\mathbf{,}}{{\mathbit{y}}}_{{\mathbf{2}}}$, and satisfy the corresponding homogeneous equation.]

The three functions ${V}_{1},{V}_{2}$ and ${V}_{3}$ that satisfy the systems are given as;

${V}_{1}^{\text{'}}{y}_{1}+{V}_{2}^{\text{'}}{y}_{2}+{V}_{3}^{\text{'}}{y}_{3}=0\phantom{\rule{0ex}{0ex}}{V}_{1}^{\text{'}}{y}_{1}^{\text{'}}+{V}_{2}^{\text{'}}{y}_{2}^{\text{'}}+{V}_{3}^{\text{'}}{y}_{3}^{\text{'}}=0\phantom{\rule{0ex}{0ex}}{V}_{1}^{\text{'}}{y}_{1}^{\text{'}\text{'}}+{V}_{2}^{\text{'}}{y}_{2}^{\text{'}\text{'}}+{V}_{3}^{\text{'}}{y}_{3}^{\text{'}\text{'}}=g.\phantom{\rule{0ex}{0ex}}$

See the step by step solution

## Step 1: Determine the three unknown function

Consider the differential equation

$y\text{'}\text{'}\text{'}\left(x\right)+{p}_{1}y\text{'}\text{'}\left(x\right)+{p}_{2}y\text{'}\left(x\right)+{p}_{3}y\left(x\right)=g\left(x\right)$

where the coefficient functions ${p}_{1},{p}_{2}$ and ${p}_{3}$ as well as $g$ are continuous on$\left(a,b\right)$ . To find a particular solution to the given equation we need to know a fundamental solution set $\left\{{y}_{1},{y}_{2},{y}_{3}\right\}$ for the corresponding homogeneous equation

$y\text{'}\text{'}\text{'}\left(x\right)+{p}_{1}y\text{'}\text{'}\left(x\right)+{p}_{2}y\text{'}\left(x\right)+{p}_{3}y\left(x\right)=0.$

Therefore, a general solution to this homogeneous equation is

${y}_{h}\left(x\right)={c}_{1}{y}_{1}\left(x\right)+{c}_{2}{y}_{2}\left(x\right)+{c}_{3}{y}_{3}\left(x\right)$

where ${c}_{1},{c}_{2}$and ${c}_{3}$ are arbitrary constants. In the method of variation of parameters, we assume there exists a particular solution to the given equation of the form

${y}_{p}\left(x\right)={V}_{1}\left(x\right){y}_{1}\left(x\right)+{V}_{2}\left(x\right){y}_{2}\left(x\right)+{V}_{3}\left(x\right){y}_{3}\left(x\right)$

and we try to determine the functions ${V}_{1}\left(x\right),{V}_{2}\left(x\right)$ and ${V}_{3}\left(x\right)$. There are three unknown functions so we will need three equations to determine them. Differentiating ${y}_{p}\left(x\right)$ gives us

${y}_{p}^{\text{'}}=\left({V}_{1}{y}_{1}^{\text{'}}+{V}_{2}{y}_{2}^{\text{'}}+{V}_{3}{y}_{3}^{\text{'}}\right)+\left({V}_{1}^{\text{'}}{y}_{1}+{V}_{2}^{\text{'}}{y}_{2}+{V}_{3}^{\text{'}}{y}_{3}\right)\text{.}$

To prevent second derivatives of the unknowns ${V}_{1},{V}_{2}$ and ${V}_{3}$ from entering the formula ${y}_{p}^{\text{'}\text{'}}$ we impose the condition

${V}_{1}^{\text{'}}{y}_{1}+{V}_{2}^{\text{'}}{y}_{2}+{V}_{3}^{\text{'}}{y}_{3}=0$

In the same manner, we impose the next condition

${V}_{1}^{\text{'}}{y}_{1}+{V}_{2}^{\text{'}}{y}_{2}+{V}_{3}^{\text{'}}{y}_{3}=0$

## Step 2: Determine the solution by using three equations.

Finally, the third condition that we impose is that ${y}_{p}$satisfies the given equation.

${y}_{p}^{\text{'}\text{'}\text{'}}\left(x\right)+{p}_{1}{y}_{p}^{\text{'}\text{'}}\left(x\right)+{p}_{2}{y}_{p}^{\text{'}}\left(x\right)+{p}_{3}{y}_{p}\left(x\right)=g\left(x\right)\phantom{\rule{0ex}{0ex}}\left({V}_{1}{y}_{1}^{\text{'}\text{'}\text{'}}+{V}_{2}{y}_{2}^{\text{'}\text{'}\text{'}}+{V}_{3}{y}_{3}^{\text{'}\text{'}\text{'}}\right)+\left({V}_{1}^{\text{'}}{y}_{1}^{\text{'}\text{'}}+{V}_{2}^{\text{'}}{y}_{2}^{\text{'}\text{'}}+{V}_{3}^{\text{'}}{y}_{3}^{\text{'}\text{'}}\right)+{p}_{1}\left({V}_{1}{y}_{1}^{\text{'}\text{'}}+{V}_{2}{y}_{2}^{\text{'}\text{'}}+{V}_{3}{y}_{3}^{\text{'}\text{'}}\right)\phantom{\rule{0ex}{0ex}}+{p}_{2}\left({V}_{1}{y}_{1}^{\text{'}}+{V}_{2}{y}_{2}^{\text{'}}+{V}_{3}{y}_{3}^{\text{'}}\right)+{p}_{3}\left({V}_{1}{y}_{1}+{V}_{2}{y}_{2}+{V}_{3}{y}_{3}\right)=g\left(x\right)\phantom{\rule{0ex}{0ex}}{V}_{1}\left({y}_{1}^{\text{'}\text{'}\text{'}}+{p}_{1}{y}_{1}^{\text{'}\text{'}}+{p}_{2}{y}_{1}^{\text{'}}+{p}_{3}{y}_{1}\right)+{V}_{2}\left({y}_{2}^{\text{'}\text{'}\text{'}}+{p}_{1}{y}_{2}^{\text{'}\text{'}}+{p}_{2}{y}_{2}^{\text{'}}+{p}_{3}{y}_{2}\right)\phantom{\rule{0ex}{0ex}}+{V}_{3}\left({y}_{3}^{\text{'}\text{'}\text{'}}+{p}_{1}{y}_{3}^{\text{'}\text{'}}+{p}_{2}{y}_{3}^{\text{'}}+{p}_{3}{y}_{3}\right)+\left({V}_{1}^{\text{'}}{y}_{1}^{\text{'}\text{'}}+{V}_{2}^{\text{'}}{y}_{2}^{\text{'}\text{'}}+{V}_{3}^{\text{'}}{y}_{3}^{\text{'}\text{'}}\right)=g\left(x\right)\phantom{\rule{0ex}{0ex}}{V}_{1}^{\text{'}}{y}_{1}^{\text{'}\text{'}}+{V}_{2}^{\text{'}}{y}_{2}^{\text{'}\text{'}}+{V}_{3}^{\text{'}}{y}_{3}^{\text{'}\text{'}}=g\left(x\right)\phantom{\rule{0ex}{0ex}}$

So, using the previous conditions and the fact that ${y}_{1},{y}_{2}$and ${y}_{3}$are solutions to the homogenous equation we get;

${V}_{1}^{\text{'}}{y}_{1}^{\text{'}\text{'}}+{V}_{2}^{\text{'}}{y}_{2}^{\text{'}\text{'}}+{V}_{3}^{\text{'}}{y}_{3}^{\text{'}\text{'}}=g.$

Therefore, we seek three functions ${V}_{1},{V}_{2}$ and ${V}_{3}$ that satisfy the system;

${V}_{1}^{\text{'}}{y}_{1}+{V}_{2}^{\text{'}}{y}_{2}+{V}_{3}^{\text{'}}{y}_{3}=0\phantom{\rule{0ex}{0ex}}{V}_{1}^{\text{'}}{y}_{1}^{\text{'}}+{V}_{2}^{\text{'}}{y}_{2}^{\text{'}}+{V}_{3}^{\text{'}}{y}_{3}^{\text{'}}=0\phantom{\rule{0ex}{0ex}}{V}_{1}^{\text{'}}{y}_{1}^{\text{'}\text{'}}+{V}_{2}^{\text{'}}{y}_{2}^{\text{'}\text{'}}+{V}_{3}^{\text{'}}{y}_{3}^{\text{'}\text{'}}=g$