Answer Happy

Series solution of variable-coefficient ODE Consider the variable coefficient linear second order homogeneous ODE (x² + 1)y" — 4xy' + 6y = 0. (1) 2 1. The point x = 0 is an ordinary point of equation (1). Therefore, we can find a power series solution of the form. y=Lamam. m=0 Write down the first and second derivatives of the power series. 2 2. Substitute the power series (and its derivatives) into equation (1). Express your answer in the form ∞ ∞ bmxm-2 + Σ Cmxm = 0, m=0 m=0 where bm and Cm are to be written in terms of m and am. 2 3. Shift the index on one of the series you found in 2 so that the exponents of x are equal to Im in both series. 2 4. Find a recurrence relation for the coefficients am+2 in terms of am and m. 4 5. Use the recurrence relation to find expressions for the coefficients a2, a3, a4 and a5. 3 6. Write down the general solution to (1) in the form y = aof (x) + a1g(x). 3 7. Find the particular solution of (1) that satisfies the initial conditions y(0) = 3 and y' (0) = 2.