PowerSeries1

PowerSeries1 - POWER SERIES Solving a second order...

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POWER SERIES Solving a second order homogeneous differential equation using the power series 17. p479 ′′ + ′ - = y xy y 2 0 y y ( ) , ( ) 0 1 0 0 = = To solve this problem we will: 1. Set y equal to a power series and substitute the power series and its derivatives into the equation for y and the derivatives of y . 2. Manipulate the expression so that x and the summation operators can be eliminated. 3. Form a recurrence relation from this expression by solving for a constant other than c n . 4. Substitute values for n into the recurrence relation to form two new series expressions involving two independent constants. This is the general solution in rough form. 5. Simplify these series as sums where possible by recognizing patterns which can be expressed as sums. 6. Substitute the first initial value into the solution and substitute the second initial value into its derivative to determine the values of the two constants. 7. Substitute the values of the two constants into the general solution to find the particular solution. let y c x n n n = = 0 then ′ = - = y nc x n n n 1 1 and ′′ = - - = y n n c x n n n ( ) 1 2 2 Substitute these expressions into the original equation n n c x x nc x c x n n n n n n n n n
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PowerSeries1 - POWER SERIES Solving a second order...

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