Transcription of 1.10 Numerical Solution to First-Order Differential Equations
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1.10 Numerical Solution to First-Order Differential Equations
I i main . 2007/2/16. page 90. i i 90 CHAPTER 1 First-Order Differential Equations 31. Consider the general rst-order linear Differential (b) Show that the general Solution to Equation equation ( ) can be written in the form dy x . + p(x)y = q(x), ( ). dx y(x) = I 1 I (t)q(t) dt + c , where p(x) and q(x) are continuous functions on some interval (a, b). where I is given in Equation ( ), and c is an arbitrary constant. (a) Rewrite Equation ( ) in Differential form, and show that an integrating factor for the result- ing equation is . I (x) = e p(x)dx . ( ). Numerical Solution to First-Order Differential Equations So far in this chapter we have investigated rst-order Differential Equations geometrically via slope elds, and analytically by trying to construct exact solutions to certain types of Differential Equations .
1, y 1) Tangent line to the solution curve passing through (x 1, y 1) Tangent line at the point (x 0, y 0) to the exact solution to the IVP (x 0, y 0) (x 1, y 1) (x 1, y(x 1)) (x 2, )) Figure 1.10.1: Euler’s method for approximating the solution to the initial-value problem dy/dx= f(x,y), y(x0) = y0. Setting x = x1 in this equation yields the ...
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