Initial value problem

Question

Determine the first 4 coefficients (a0, a1, a2and a3) of a series solution around to=0 for the initial value problem below

(d/dt y(t)) +(-2t - 1) y(t) = 0, y(0) = −1

Tip: express the solution in the form:

y(t) = ao+a1t+ a₂t² + a3t³ + ...

Answer 1

−There is a solution to this beautiful problem using summation symbols ∑ .

I am unable to present it here because of the luck of the lower and upper limits

in the summation symbols of the existing editor.

Using the aforementioned method we ended up a₀ = a₁= -1

and the recursive formula a_n+1=( 2 a_n-1+a_n ) / (n + 1) for all n≥ 1

For n=1 the recursive formula yields a₂= -3/2

For n =2 we get a₃= -7/6 and so on.

Another approach , would be to notice that the given differential equation is a separable one

with solution y(t) = −e^{t^2} ·e^t . Then expand e^{t^2} and e^t as power series , then multiply them and

collect the four first terms.