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The Chemical Recycling Of Polyesters For A Circular Plastics Economy Challenges And Emerging Opportunities Payne 21 Chemsuschem Wiley Online Library

I Î æ B 1 m B è H 0 I Î æ @ e ô 1 @ ) / ÷ { H × H K § k m E > @ I Ú !G } b U G& ö l g 0d ö b0{!l ?Must be solved to get u= g(x,y,c 2) on characteristics λ(x,y) = c 1, where c 2 is another constant of integration 3 The two arbitrary constants c 1 and c 2 can be thought of as being related by an arbitrary function c 2 = f c 1 12 Seven examples Example 1 Consider the simple PDE u x u

Ï X Æ Æ Ã Æ ® Ê S j Å ¸ Ò R 0 !)Kptfqi Ulbdi* ú x â ý _ b)Usbot!gpsnfe!cz!Usvui* I ¡ w > } û À Ü } 8 @ ³/!!½ Q C ¾ ¼ P Ð _ Ì ¸ g Ì Q l ú 6 Æ Â Ì « ê ç U J Ï 6 ½ Q C ¾ ¼ P Ð _ Ì ¸ g Ì Q l ú 6 Æ Â Ì « ê ç U J Ï 6Since f(−x) = e− (− x) 2 2 = e− 2 = f(x) and lim x→±∞ e− (−x)2 2 = 0, the graph is symmetry wrt the yaxis, and the xaxis is a horizontal asymptote • Wehave f0(x) = e−x 2 2 (−x) = −xe− x2 2 • Thus f ↑ on (−∞,0) and ↓ on (0,∞)

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