310_pdfsam_math 54 differential equation solutions odd

310_pdfsam_math 54 differential equation solutions odd -...

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Chapter 5 (b) The condition v 6 = 0 corresponds to the third case in (5.28), i.e., the system has the form y 0 = v, v 0 = y sign( v ) . The phase plane equation for this system is dv dy = dv/dt dy/dt = y sign( v ) v . We consider two cases. 1) v> 0. In this case sign( v ) = 1 and we have dv dy = y 1 v vdv = ( y +1) dy Z vdv = Z ( y +1) dy 1 2 v 2 = 1 2 ( y +1) 2 + C v 2 +( y +1) 2 = c, where c =2 C . 2) v< 0. In this case sign( v )= 1 and we have dv dy = y +1 v vdv = ( y 1) dy Z vdv = Z ( y 1) dy 1 2 v 2 = 1 2 ( y 1) 2 + C v 2 +( y 1) 2 = c. (c) The equation v 2 +( y +1) 2 = c deFnes a circle in the yv -plane centered at ( 1 , 0) and of the radius
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Unformatted text preview: √ c if c > 0, and it is the empty set if c < 0. The condition v > 0 means that we have to take only the half of these circles lying in the upper half plane. Moreover, the Frst equation, y = v , implies that trajectories ﬂow from left to right. Similarly, in the lower half plane, v < 0, we have concentric semicircles v 2 + ( y − 1) 2 = c , c ≥ 0, centered at (1 , 0) and ﬂowing from right to left. 306...
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at Berkeley.

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