Explanation Given information: The given u ( x , t ) = sech 2 ( x − t ) Proof: The given function, u ( x , t ) = sech 2 ( x − t ) We have to prove 4 u t + u x x x + 12 u u x = 0 Consider, u ( x , t ) = sech 2 ( x − t ) Differentiating both sides with respect to t , we get, u t = ∂ ∂ t sech 2 ( x − t ) ⇒ u t = 2 sech ( x − t ) [ − sech ( x − t ) tanh ( x − t ) ] ∂ ∂ t ( x − t ) ⇒ u t = 2 sech 2 ( x − t ) tanh ( x − t ) … ( 1 ) Now differentiating u ( x , t ) = sech 2 ( x − t ) with respect to x , we get u x = ∂ ∂ x sech 2 ( x − t ) u x = 2 sech ( x − t ) [ − sech ( x − t ) tanh ( x − t ) ] ∂ ∂ t ( x − t ) ⇒ u x = − 2 sech 2 ( x − t ) tanh ( x − t ) … ( 2 ) Again differentiating both sides we get, u x x = ∂ ∂ x [ − 2 sech 2 ( x − t ) tanh ( x − t ) ] ⇒ u x x = − 2 [ − 2 sech 2 ( x − t ) tanh ( x − t ) tanh ( x − t ) + sech 4 ( x − t ) ] ⇒ u x x = 4 sech 2 ( x − t ) tanh 2 ( x − t ) − 2 sech 4 ( x − t ) Differentiating both sides with respect x we get, u x x x = ∂ ∂ x [ 4 sech 2 ( x − t ) tanh 2 ( x − t ) − 2 sech 4 ( x − t ) ] ⇒ u x x x = 4 [ − 2 sech 2 ( x − t ) tanh 3 ( x − t ) + 2 tanh ( x − t ) sech 4 ( x − t ) ] + 8 sech 4 ( x − t ) tanh ( x − t ) … ( 3 ) Now we have to show 4 u t + u x x x + 12 u u x = 0 Hence

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Chapter 14.3, Problem 79E

To determine

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that u(x,t)=sech2(x−t) satisfies the Korteweg-de vries equation (which arises in the study of water waves): 4ut+uxxx+12uux=0

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Chapter 14.3, Problem 78E

Chapter 14.3, Problem 80E

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Chapter 14 Solutions

EBK CALCULUS: EARLY TRANSCENDENTALS

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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Toshow: that u ( x , t ) = sech 2 ( x − t ) satisfies the Korteweg-de vries equation (which arises in the study of water waves): 4 u t + u x x x + 12 u u x = 0

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Toshow: that u(x,t)=sech2(x−t) satisfies the Korteweg-de vries equation (which arises in the study of water waves): 4ut+uxxx+12uux=0

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Chapter 14 Solutions

Toshow:

that u(x,t)=sech2(x−t) satisfies the Korteweg-de vries equation (which arises in the study of water waves): 4ut+uxxx+12uux=0