Answer Happy

eries solution of variable-coefficient ODE Consider the variable coefficient linear second order homogeneous ODE (r² + 1)g" - Ary + 6y = 0. (1) 1. The point z = 0 is an ordinary point of equation (1). Therefore, we can find a power series solution of the form. DXC y=Σama". 70 Write down the first and second derivatives of the power series. 2. Substitute the power series (and its derivatives) into equation (1). Express your answer in the form EXO 2 TH 70 771( where b and care to be written in terms of m and am 3. Shift the index on one of the series you found in 2 so that the exponents of rare equal to m in both series. 4. Find a recurrence relation for the coefficients a 2 in terms of an and m. 5. Use the recurrence relation to find expressions for the coefficients a2, 03, 04 and as. 6. Write down the general solution to (1) in the form y = aof(z) + a,g(x). 7. Find the particular solution of (1) that satisfies the initial conditions y(0) = 3 and y'(0) = 2.