Use Power Series to solve this equation. dx/dt + (t^2)x = 0. x = a0 + a1t + a2(t^2) + ... + an(t^n) x = ∞∑(n=0) an(t^n) dx/dt = a1 + 2(a2)t + 3(a3)(t^2) + ... + n(an)(t^n) dx/dt = ∞∑(n=1) n(an)(t^(n-1)) (=) ∞∑(n=1) n(an)(t^(n-1)) + (t^2)[∞∑(n=0) an(t^n)] = 0 ∞∑(n=1) n(an)(t^(n-1)) = ∞∑(n=0) (n+1)(a)(t^n) (t^2)[∞∑(n=0) an(t^n)] = ∞∑(n=2) (a)(t^n) (=) ∞∑(n=0) (n+1)(a)(t^n) + ∞∑(n=2) (a)(t^n) = 0 a1 + a2t + ∞∑(n=2) (n+1)(a)(t^n) + ∞∑(n=2) (a)(t^n) = 0 a1 + a2t + ∞∑(n=2) [(n+1)(a) + (a)] (t^n) = 0 my lecturer: what is a0, a1, a2, ..., an?

Question

Use Power Series to solve this equation. dx/dt + (t^2)x = 0. x = a0 + a1t + a2(t^2) + ... + an(t^n) x = ∞∑(n=0) an(t^n) dx/dt = a1 + 2(a2)t + 3(a3)(t^2) + ... + n(an)(t^n) dx/dt = ∞∑(n=1) n(an)(t^(n-1)) (=) ∞∑(n=1) n(an)(t^(n-1)) + (t^2)[∞∑(n=0) an(t^n)] = 0 > ∞∑(n=1) n(an)(t^(n-1)) = ∞∑(n=0) (n+1)(a)(t^n) > (t^2)[∞∑(n=0) an(t^n)] = ∞∑(n=2) (a)(t^n) (=) ∞∑(n=0) (n+1)(a)(t^n) + ∞∑(n=2) (a)(t^n) = 0 a1 + a2t + ∞∑(n=2) (n+1)(a)(t^n) + ∞∑(n=2) (a)(t^n) = 0 a1 + a2t + ∞∑(n=2) [(n+1)(a) + (a)] (t^n) = 0 my lecturer: what is a0, a1, a2, ..., an?