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11_pdfsam_math 54 differential equation solutions odd

11_pdfsam_math 54 differential equation solutions odd -...

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Exercises 1.2 –10 10 –0.5 0.5 C =2 C =1 C =0 . 5 C =0 C = 0 . 5 C = 1 C = 2 Figure 1–A : Graphs of the functions y = Ce 3 x + 1 for C = 0, ± 0 . 5, ± 1, and ± 2. 17. Differentiating φ ( x ), we find that φ ( x ) = 2 1 ce x = 2 (1 ce x ) 1 = 2( 1) (1 ce x ) 2 (1 ce x ) = 2 ce x (1 ce x ) 2 . (1.2) On the other hand, substitution of φ ( x ) for y into the right-hand side of the given equation yields φ ( x )( φ ( x ) 2) 2 = 1 2 2 1 ce x 2 1 ce x 2 = 2 1 ce x 1 1 ce x 1 = 2 1 ce x 1 (1 ce x ) 1 ce x = 2 ce x (1 ce x ) 2 , which is identical to φ ( x ) found in (1.2).
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