Find at least the first four nonzero terms in a power series expansion about x=0 for a general solution to the given differential equation. y" +(x-7)y' +y=0 y(x) = +... (Type an expression in terms of ag and a that includes all terms up to order 3.)

Consider the following differential equation and initial value. y' = 2x - 3y + 1, y(1) = 6; y(1.2) Use Euler's method to obtain a four-decimal approximation of the indicated value. Carry out the recursion of (3) in Section 2.6 First, use increment h = 0.1 1.1 Yn + 1 = Yn + hf(xnr Yn) 1.05 1.15 y(1) y(1.2) y(1) y(1.1) 5.99 Then, use increment h = 0.05. (Round your answers to four decimal places.) y(1.2) 5.87 5.9975 5.9425 5.885 6 5.875 (3) 6 X X X X X

The point x = O is a regular singular point of the given differential equation. Find the recursive relation for the series solution of the DE below. Show the substitution and all the steps to obtain the recursive relation. Do not solve the equation for y=y(x) xy" + 4y' - xy = 0, 1 a. Ck+1= (k+r+ 1)(k+r+3) 1 b. Ck+1 = (k+r+1)(k+r+4) ·Ck-1, k≥ 1 с. Ск d. Ck² e. Ck O a P = e 11 || 1 k+r I k+r ·Ck-1, k≥1 -Ck-1, k≥1 -Ck-1, k≥1 (k+r)² +5(k+r) 1 (k+r)²-2(k+r) -8 ·CK-2, k≥2

Find a power series expansion about x = 0 for a general solution to the given differential equation. Your answer should include a general formula for the coefficients. y' - 2xy = 0 00 y=ao Σ ( n=0