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Homework-1-s11-solution

# Homework-1-s11-solution - AMS 361 Applied Calculus IV...

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1 AMS 361: Applied Calculus IV Homework 1 Spring 2011 Problem 1-1: Verify by substitution that the given functions are solutions of the given differential equation. Primes denote derivatives with respect to x . 𝑥 2 𝑦 ′′ + 𝑥𝑦 − 𝑦 = ln 𝑥 ; 𝑦 1 = 𝑥 − ln 𝑥 , 𝑦 2 = 1 𝑥 ln 𝑥 Solution: First take all the necessary derivatives, then substitute accordingly. 𝑦 1 = x ln 𝑥 𝑦 2 = 1 𝑥 ln 𝑥 𝑦 1 = 1 1 𝑥 𝑦 2 = 1 𝑥 2 1 𝑥 𝑦 1 ′′ = 1 𝑥 2 𝑦 2 ′′ = 1 𝑥 3 + 1 𝑥 2 Problem 1-2: Verify that y(x) satisfies the given differential equation and then determine a value of the constant c so that y(x) satisfies the given initial condition. 𝑒 𝑦 𝑦 = 1; 𝑦 ( 𝑥 ) = ln( 𝑥 + 𝐶 ) , 𝑦 (0) = 0 Solution: First we compute the derivative, 𝑦 ( 𝑥 ) = ln( 𝑥 + 𝐶 ) 𝑦 ( 𝑥 ) = 1 𝑥 + 𝐶 Next we substitute accordingly, 𝑒 𝑦 ( 𝑥 ) 𝑦 ( 𝑥 ) = 𝑒 ln ( 𝑥+𝐶 ) 1 𝑥 + 𝐶 = ( 𝑥 + 𝐶 ) 1 𝑥 + 𝐶 = 1

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Homework-1-s11-solution - AMS 361 Applied Calculus IV...

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