Solving Differential Equations by the Laplace （解决由拉普拉斯微分方程）.pdfVIP

36CH_PHCalter_TechMath_950299 23/2/2007 2:28 PM Page 1 Solving Differential Equations by the Laplace Transform and by Numerical Methods OBJECTIVES When you have completed this chapter, you should be able to: • Find the Laplace transform of a function by direct integration or by using a table. • Find the Laplace transform of an expression containing derivatives, and substitute initial conditions. • Determine the inverse of a Laplace transform by completing the square or by partial fractions. • Solve first-order and second-order differential equations using Laplace transforms. • Solve electrical applications using Laplace transforms. • Solve first-order differential equations using Euler’s method, the modi- fied Euler’s method, or the Runge-Kutta method. • Solve second-order differential equations using the modified Euler’s method or the Runge-Kutta method. The methods for solving differential equations that we learned in Chapters 28 and 29 are often called classical methods. In this chapter we learn other powerful tech- niques: the Laplace transform and numerical methods. The Laplace transform en- ables us to transform a differential equation into an algebraic one. We can then solve the algebraic equation and apply an inverse transform to obtain the solution to the differential equation. The Laplace transform is good for finding particular solutions of differential equations. For example, it enables us to solve initial v