M427K-F10-HW3 - x ) = xe x . 3.4.25 Use the method of...

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M427K , 55330, Sep 13, 2010 Homework 3 , due Sep 20 2.8.10a Let φ 0 ( t ) = 0 and use the method of successive approximations (Picard’s iteration method) to solve the given initial value problem: Calculate φ 1 ( t ), φ 2 ( t ), and φ 3 ( t ). y = 1 - y 3 , y (0) = 0 . 3.2.14 Verify that y 1 ( t ) = 1 and y 2 ( t ) = t 1 / 2 are solutions of the diFerential equation yy ′′ + ( y ) 2 = 0 for t > 0. Then show that y = c 1 + c 2 t 1 / 2 is not, in general, a solution of this equation. Explain why this result does not contradict Theorem 3.2.2. 3.2.26 Verify that the functions y 1 and y 2 are solutions of the given diFerential equation. Do they constitute a fundamental set of solutions? x 2 y ′′ - x ( x + 2) y + ( x + 2) y = 0 , x > 0; y 1 ( x ) = x, y 2 (
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Unformatted text preview: x ) = xe x . 3.4.25 Use the method of reduction of order to nd a second solution of the given diFerential equation. t 2 y + 3 ty + y = 0 , t > 0; y 1 ( t ) = t 1 . 3.4.26 Use the method of reduction of order to nd a second solution of the given diFerential equation. t 2 y -t ( t + 2) y + ( t + 2) y = 0 , t > 0; y 1 ( t ) = t. 3.4.30 Use the method of reduction of order to nd a second solution of the given diFerential equation. x 2 y + xy + ( x 2-1 4 ) y = 0 , x > , y 1 ( x ) = x 1 / 2 sin( x ) ....
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