题目

MAP2302 Practice Final Exam

多项填空题

Find the first four nonzero terms in a power series expansion about x_0=4x_0=4 for the differential equation given below. \left(x^2-8x\right)y''+3y=0\left(x^2-8x\right)y''+3y=0. y\left(x\right)=a_0\left[a+b\left(x-4\right)^2+...\right]+a_1\left[c\left(x-4\right)+d\left(x-4\right)^3+...\right]y\left(x\right)=a_0\left[a+b\left(x-4\right)^2+...\right]+a_1\left[c\left(x-4\right)+d\left(x-4\right)^3+...\right] a=a= [Fill in the blank], b=b= [Fill in the blank], c=c= [Fill in the blank], d=d= [Fill in the blank], (enter integers or fractions)

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思路分析

We begin by restating the problem in a form that's convenient for a series expansion. The differential equation is (x^2 - 8x) y'' + 3 y = 0, and we expand about x0 = 4. Let t = x - 4, so x = t + 4. Then x^2 - 8x becomes (t+4)^2 - 8(t+4) = t^2 - 16. Thus the equation becomes (t^2 - 16) y'' + 3 y = 0, with derivatives taken with respect to t (which matches derivatives with respect to x since t = x - 4 simply shifts the origin). We are given a structured form for the solution: y(t) = a0 [ a + b t^2 + ... ] + a1 [ c t + d t^3 + ... ]. Here a, b, c, d are constants to be determined (other higher-order coefficients exist but are not needed for the first four nonzero terms). We will substitute this form into the differential equation and equate coefficients of like powers of t to obtain......Login to view full explanation