26_pdfsam_math 54 differential equation solutions odd

26_pdfsam_math 54 differential equation solutions odd - 11...

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Chapter 1 –1 0 1.2 1.4 1.6 1.8 2 Polygonal approximation y = 1 /x Figure 1–C : Polygonal line approximation and the actual solution for Problem 9. we compute both sides of the given diFerential equation: y 0 ( x )= ( x 1 ) 0 = x 2 , f ( x, y ( x )) = x 2 ( x 1 ) x 1 ( x 1 ) 2 = x 2 + x 2 x 2 = x 2 . Thus, the function y ( x )= 1 /x is, indeed, the solution to the given initial value problem. The graphs of the obtained polygonal line approximation and the actual solution are sketched in ±igure 1-C.
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Unformatted text preview: 11. In this problem, the independent variable is t and the dependent variable is x ; f ( t, x ) = 1+ x 2 , t = 0, and x = 0. The function φ ( t ) = tan t satis²es the initial condition: φ (0) = tan 0 = 0. The diFerential equation is also satis²ed: dφ dt = sec 2 t = 1 + tan 2 t = 1 + φ ( t ) 2 . Therefore, φ ( t ) is the solution to the given initial value problem. 22...
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