127_pdfsam_math 54 differential equation solutions odd

127_pdfsam_math 54 differential equation solutions odd - dT...

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Exercises 3.4 After 12 hours of sunlight, the temperature will be T (12) = 336 226 e 12 / 64 148 . 6 F . 15. The equation dT/dt = k ( M 4 T 4 ) is separable. Separation variables yields dT T 4 M 4 = kdt Z dT T 4 M 4 = Z kdt = kt + C 1 . (3.9) Since T 4 M 4 =( T 2 M 2 )( T 2 + M 2 ), we have 1 T 4 M 4 = 1 2 M 2 ( M 2 + T 2 )+( M 2 T 2 ) ( T 2 M 2 )( T 2 + M 2 ) = 1 2 M 2 ± 1 T 2 M 2 1 T 2 + M 2 ² , and the integral in the left-hand side of (3.9) becomes Z dT T 4 M 4 = 1 2 M 2 ±Z dT T 2 M 2 Z dT T 2 + M 2 ² = 1 4 M 3 ± ln T M T + M 2 arctan ³ T M ´² . Thus a general solution to Stefan’s equation is given implicitly by 1 4 M 3 ± ln T M T + M 2 arctan ³ T M ´² = kt + C 1 or T M = C ( T + M )exp ± 2 arctan ³ T M ´ 4 M 3 kt ² . When T is close to M , M 4 T 4 =( M T )( M + T ) ( M 2 + T 2 ) ( M T )(2 M ) ( 2 M 2 ) 4 M 3 ( M T ) , and so
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Unformatted text preview: dT dt k 4 M 3 ( M T )4 M 3 = k 1 ( M T ) with k 1 = 4 M 3 k , which constitutes Newtons law. EXERCISES 3.4: Newtonian Mechanics, page 115 1. This problem is a particular case of Example 1 on page 110 of the text. Therefore, we can use the general formula (6) on page 111 with m = 5, b = 50, and v = v (0) = 0. But let us follow the general idea of Section 3.4, nd an equation of the motion, and solve it. 123...
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