Consider the variable coefficient linear second order homogeneous ODE y (x) xy' (x) y(x)...

 

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Consider the variable coefficient linear second order homogeneous ODE y" (x) — xy' (x) − y(x) = 0. 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 = Σ amxm 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 Σbm 2² m bmx™-2 + Σ cmam = 0, m=0 m=0 where bm and cm are to be written in terms of m and am. (1) 2 3. Shift the index on one of the series you found in 2 so that the exponents of x are equal to m 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 in terms of ao and a₁. 2 6. Write down the general solution to (1) in the form y = aof (x) + a1g(x). 2 7. Determine the values of ao and a₁ for the particular solution satisfying the initial conditions y(0) = 1 and y'(0) = 0. 2 8. Deduce that the solution in 7 can be expressed as an elementary function. Hint: un e² = Σ n! n=0 Consider the variable coefficient linear second order homogeneous ODE y" (x) — xy' (x) − y(x) = 0. 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 = Σ amxm 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 Σbm 2² m bmx™-2 + Σ cmam = 0, m=0 m=0 where bm and cm are to be written in terms of m and am. (1) 2 3. Shift the index on one of the series you found in 2 so that the exponents of x are equal to m 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 in terms of ao and a₁. 2 6. Write down the general solution to (1) in the form y = aof (x) + a1g(x). 2 7. Determine the values of ao and a₁ for the particular solution satisfying the initial conditions y(0) = 1 and y'(0) = 0. 2 8. Deduce that the solution in 7 can be expressed as an elementary function. Hint: un e² = Σ n! n=0