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12_pdfsam_math 54 differential equation solutions odd

# 12_pdfsam_math 54 differential equation solutions odd -...

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Chapter 1 21. For φ ( x ) = x m , we have φ ( x ) = mx m 1 and φ ( x ) = m ( m 1) x m 2 . (a) Substituting these expressions into the differential equation, 3 x 2 y + 11 xy 3 y = 0, gives 3 x 2 m ( m 1) x m 2 + 11 x mx m 1 3 x m = 0 3 m ( m 1) x m + 11 mx m 3 x m = 0 [3 m ( m 1) + 11 m 3] x m = 0 3 m 2 + 8 m 3 x m = 0 . For the last equation to hold on an interval for x , we must have 3 m 2 + 8 m 3 = (3 m 1)( m + 3) = 0 . Thus either (3 m 1) = 0 or ( m + 3) = 0, which gives m = 1 3 , 3. (b) Substituting the above expressions for φ ( x ), φ ( x ), and φ ( x ) into the differential equa- tion, x 2 y xy 5 y = 0, gives x 2 m ( m 1) x m 2 x mx m 1 5 x m = 0 m 2 2 m 5 x m = 0 . For the last equation to hold on an interval for
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