Find the indicated coefficients of the power series solution about x=0 of the differential equation

Write the ODE as



and truncate as many terms as needed to obtain only the terms up to order 4:




We only care about the coefficients up to , so we take the system



Now, we're given initial values and , so that




which gives

2.4+3.2t. you get the answer by adding the ones with variables and ones without