1_1 - Solutions 1.1-Page 9 Problem 1 Verify by substitution...

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Solutions 1.1-Page 9 Problem 1 Verify by substitution that each given function is a solution of the given differential equation. Primes denote derivatives with respect to x. 0 2 = + y y ; x e y 2 3 = Eq.1: Eq.2: y 3 = 0 2 = + y y x e 2 Differentiating Eq.2 with respect to x yields: a) x e y 2 6 = From Eq.2, b) 2 x e y 2 6 = Substituting a and b into Eq.1 yields: 0 6 6 2 2 = + x x e e
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Problem 6 Verify by substitution that each given function is a solution of the given differential equation. Primes denote derivatives with respect to x. 0 4 4 = + + y y y ; x x xe y e y 2 2 2 1 , = = The problem will first be verified for , and then the problem will be verified again for . 1 y 2 y With : 1 y Eq.1: Eq.2: e y = 0 4 4 = + + y y y x 2 1 Differentiating Eq.2 with respect to x and multiplying by 4 yields: a) x e y 2 1 8 4 = Differentiating again with respect to x yields: b) x e y 2 1 4 = Multiplying Eq.2 by 4 yields: c) x e y 2 1 4 4 = Substituting a, b, and c into Eq.1 yields: 0 4
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