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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 4: Q52 E (page 201)

The reduction of order formula $(13)$ can also be derived from Abel鈥檚 identity (Problem 32). Let $f (t)$ be a nontrivial solution toanda second linearly independent solution. Show that ${(\frac{y}{f})}^{'} = \frac{W [f, y]}{f^{2}}$ and then use Abel's identity for the Wronskian $W [f, y]$ to obtain the reduction of order formula.

Short Answer

Expert verified

The solution of the given equation is $y = f 鈭� \frac{- e^{鈭� p (t) dt}}{f^{2}} dt$ .

Step by step solution

01

Using the formula

From the given ${(\frac{y}{f})}^{'} = \frac{fy' - f' y}{f^{2}}$ and we know $w [f, y] = fy' - f' y$

${(\frac{y}{f})}^{'} = \frac{W [f, y]}{f^{2}}$

Hence proved.

02

Integration

Now integrate on both sides;

$鈭� {(\frac{y}{f})}^{'} dt = 鈭� \frac{W [f, y]}{f^{2}} dt \frac{y}{f} = 鈭� \frac{- e^{鈭� p (t) dt}}{f^{2}} dt y = f 鈭� \frac{- e^{鈭� p (t) dt}}{f^{2}} dt$

role="math" $鈭� {(\frac{y}{f})}^{'} dt = 鈭� \frac{W [f, y]}{f^{2}} dt \frac{y}{f} = 鈭� \frac{- e^{鈭� p (t) dt}}{f^{2}} dt y = f 鈭� \frac{- e^{鈭� p (t) dt}}{f^{2}} dt$

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Most popular questions from this chapter

Solve the given initial value problem $y'' + 2 y' + 17 y = 0;$ $y (0) = 1,$ $y' (0) = - 1$ .

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