Find at least the first 4 nonzero terms of the general solut

Find at least the first 4 nonzero terms of the general solution for y\' - 2xy = 0 near x_0 = -1. Find at least the first 4 nonzero terms of the solution to y\" + ty\' + e^t y = 0, y(0) = 0, y\'(0)= -1.

Solution

y^\'=2xy

dy/dx=2xy

dy/y=2xdx

integrating onboth sides

log y = x²+c

y=e^x²+c

y= 1+x²+x⁴/2!+x⁶/3!+........+c

near x₀=-1

y=1+1/2+1/6+....