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TAM310hw05 - it has a repeated root c Continue the method...

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TAM 310 Advanced Engineering Analysis I Spring 2008 Prof.R.Rand Homework No.5 (Due Tuesday Feb.26) 1. Determine whether x = 0 is an ordinary point, a regular singular point, or an irregular singu- lar point for each of the following. In each case, use Rules 1 thru 4 in section 8 of the notes to say something about the expected form of the series solution a. x y primeprime + ( x - x 3 ) y prime + y sin x = 0 b. x (1 + x ) y primeprime + 2 y prime + 3 x y = 0 c. 3 x 3 y primeprime + 2 x 2 y prime + (1 - x 2 ) y = 0 2. Find all singular points in the following equations and determine whether each one is regular or irregular: a. (1 - x 2 ) y primeprime - 2 x y prime + 12 y = 0 b. ( x 2 - 9) 2 y primeprime + ( x 2 + 9) y prime + ( x 2 + 4) y = 0 c. x 3 (1 - x ) y primeprime + (3 x + 2) y prime + x y = 0 3. This question concerns the ODE: ( x 3 + 3 x 2 + x ) y primeprime + (1 - x 2 ) y prime + ( - 1 + x ) y = 0 EQ.(*) a. Show that x = 0 is a regular singular point. b. Compute the indicial equation which determines
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Unformatted text preview: it has a repeated root. c. Continue the method of Frobenius to ±nd a solution y = f ( x ). HINT: The in±nite series for f ( x ) truncates to a polynomial. d. Rule 4 on page 15 of the notes applies here and says that a second linearly independent solution can be obtained in the form: y = f ( x ) log( x ) + g ( x ) Having found f ( x ) in part c above, substitute this expression into EQ.(*) and obtain an ODE on g ( x ). Solve it by looking for g ( x ) in the form of a Frobenius series. HINT: The in±nite series for g ( x ) truncates to a polynomial. e. Combine the above results to write the general solution of EQ.(*). Check your answer by substitution....
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  • Spring '08
  • PHOENIX
  • ORDINARY DIFFERENTIAL EQUATIONS, Complex differential equation, Frobenius method, Regular singular point, a. b. c., irregular singular point

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