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 . ° 2 ± ′′ + °± − ± = ln ° ; ± 1 = ° − ln ° , ± 2 = 1 ° ln ° Solution: First take all the necessary derivatives, then substitute accordingly. ± 1 = x ln ° ± 2 = 1 ² ln ° ± 1 = 1 1 ² ± 2 ³ = 1 ² ´ 1 ² ± 1 ′′ = 1 ² ´ ± 2 ³³ = 1 ² 3 + 1 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. µ ± = 1; ± ( ° ) = ln( ° + · ) , ± (0) = 0 Solution: First we compute the derivative, ± ( ° ) = ln( ° + · ) ± ( ° ) = 1 ° + · Next we substitute accordingly, µ ( ² ) ± ( ° ) = µ ln ( ²+¸ ) 1 ° + · ¹ = ( ° + · ) 1 ° + · ¹ = 1 Finally we find the value for C: ± (0) = 0 = ln(C) C = 1 Problem 1-3: Find a function () y fx = satisfying the given differential equation and the prescribed initial condition.
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This note was uploaded on 05/09/2011 for the course AMS 361 taught by Professor Staff during the Spring '08 term at SUNY Stony Brook.

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

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