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Y y y A solution is a function f x such that the substitution y f x y f x y f x gives an identity The differential equation is said to be linear if it is linear in the variables y y y We have already seen (in section 64) how to solve first order linear equations;. Æ ¿ À ¯ ± ½ ° Á É Â º ´ ³ Ã Ç ² ¼ Ð Ö » Ê Ô ° À ± µ » ² ¯ ¾ Æ º Ì ´ Â É ¼ Ã Ç Ð Á ½ ¿ Ö ³ Ñ Ò Ê Ó ¿ µ ± ´ ¯ À Æ º ° Ú ³ Ã ¼ â ® Û » × Ø Ê Ó Á ± µ ¯ ¿ Æ ° Ú Ã Þ º ´ ¼ ® ³ À × Ï Ê Þ º Æ µ ´ ¼ Ú ± À ¯ Ý ¿ ° ® ³ Å ¾ Ã » × Ù Ê Ú Ã. Proof lnexy = xy = lnex lney = ln(ex ·ey) Since lnx is onetoone, then exy = ex ·ey 1 = e0 = ex(−x) = ex ·e−x ⇒ e−x = 1 ex ex−y = ex(−y) = ex ·e−y = ex · 1 ey ex ey • For r = m ∈ N, emx = e z }m { x···x = z }m { ex ···ex = (ex)m • For r = 1 n, n ∈ N and n 6= 0, ex = e n n x = e 1 nx n ⇒ e n x = (ex) 1 • For r rational, let r = m n, m, n ∈ N.

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