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 ,wehave φ 0 ( x )= mx m 1 and φ 0 ( x )= m ( m 1) x m 2 . (a) Substituting these expressions into the diferential equation, 3 x 2 y 0 +11 xy 0 3 y =0 , gives 3 x 2 ± m ( m 1) x m 2 ² +11 x ± mx m 1 ² 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 ² x m =0 . For the last equation to hold on an interval ±or x ,wemusthave 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 ±or φ ( x ), φ 0 ( x ), and φ 0 ( x ) into the diferential equa- tion, x 2 y 0 xy 0 5 y =0,gives x 2 ± m ( m 1) x m 2 ² x ± mx m 1 ² 5 x m =0 ± m 2 2 m 5 ² x m =0 . For the last equation to hold on an interval ±or x
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