# soln11 - ECE320 Solution Notes 11 Cornell University Spring...

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ECE320 Solution Notes 11 Spring 2006 Cornell University T.L.Fine 1. Classify the following ode s as to whether they are autonomous and what their orders and degrees are: ( a ) ¨ x + ω 2 x = 0 , harmonic oscillator; ( b ) ¨ x - tx = 0 , Airy ode; ( c ) t 2 ¨ x + t ˙ x + ( λ 2 t 2 - n 2 ) x = 0 , Bessel’s equation; ( d ) ˙ x - tx 2 = e - t ; ( e ) tan(¨ x ) - x = 0; ( f ) ¨ x ˙ xx = sin( t ); ( g ) ( ˙ x ) 2 - x = 0; ( h ) ¨ x = - x 2 + sin( ω ˙ x ) . 1. autonomous, order 2, degree 1. 2. nonautonomous, order 2, degree 1. 3. nonautonomous, order 2, degree 1. 4. nonautonomous, order 1, degree1. 5. autonomous, order 2, degree infinity. 6. autonomous, order 2, degree 1. 7. autonomous, order 1, degree 2. 8. autonomous, order 2, degree 1. 2. Represent each of the ode s of Problem 1 in the standard form ˙ y = F ( y ), if possible. ( a ) y 1 = x, y 2 = ˙ x, ˙ y 1 = y 2 , ˙ y 2 = - ω 2 y 1 ; ( b ) y 0 = t, y 1 = x, y 2 = ˙ x, ˙ y 0 = 1 , ˙ y 1 = y 2 , ˙ y 2 = y 0 y 1 ; ( c ) y 0 = t, y 1 = x, y 2 = ˙ y 1 , ˙ y 0 = 1 , ˙ y 1 = y 2 , ˙ y 2 = - 1 t y 2 - ( λ 2 - n 2 t 2 ) y 1 ; ( d ) y 0 = t, y 1 = x, ˙ y 0 = 1 , ˙ y 1 = y 0 y 2 1 + e - y 0 ; ( e ) infinite degree, cannot be represented; ( f ) y 0 = t, y 1 = x, y 2 = ˙ y 1 , ˙ y 0 = 1 , ˙ y 1 = y 2 , ˙ y 2 = sin(

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• Spring '06
• FINE
• y1, Cornell University, Equilibrium point, Stability theory, -1 y1

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