Derive the taylor series for sin2t by applying the picard method to the first-order system corresponding to the second-order ivp x"=-4x ; x(0)=0; x'(0)=2

You're certainly right about reducing this second order linear ODE into a system of first order linear ODEs. I'll use different symbols to avoid any confusion with subscripts as part of the iterative process. Setting and gives and . Your initial conditions can then be written as and . The system is



The Picard process will involve the recurrence relation



First step:











Second step:











Third step:







Fourth step:







One more step for good measure:




We don't actually need unless you want to continue looking for , but I don't think we'll need to. We have enough of a pattern to find , which is the solution to the ODE because .





which is indeed the Taylor series for .