Solve-the-ODE-y-xy-x-2-with-y-0-2-Mastermind-

Question Number 169050 by Mastermind last updated on 23/Apr/22

S o l v e t h e O D E

y^{'} + x y = x^{2}, w i t h y (0) = 2

M a s t e r m i n d

Answered by Mathspace last updated on 23/Apr/22

h \to y^{^{'}} = - x y \Rightarrow \frac{y^{^{'}}}{y} = - x \Rightarrow

l n ∣ y ∣= - \frac{x^{2}}{2} + λ \Rightarrow y = k e^{- \frac{x^{2}}{2}}

(m v c) \to y^{^{'}} = k^{^{'}} e^{- \frac{x^{2}}{2}} - k x e^{- \frac{x^{2}}{2}}

(e) \Rightarrow k^{^{'}} e^{- \frac{x^{2}}{2}} - k x e^{- \frac{x^{2}}{2}} + x k e^{- \frac{x^{2}}{2}} = x^{2}

\Rightarrow k^{^{'}} = x^{2} e^{\frac{x^{2}}{2}} \Rightarrow k = \int_{0}^{x} t^{2} e^{\frac{t^{2}}{2}} d t + λ

y (x) = (λ + \int_{0}^{x} t^{2} e^{\frac{t^{2}}{2}} d t) e^{- \frac{x^{2}}{2}}

y (0) = 2 \Rightarrow λ = 2 \Rightarrow

y (x) = 2 e^{- \frac{x^{2}}{2}} + e^{- \frac{x^{2}}{2}} \int_{0}^{x} t^{2} e^{\frac{t^{2}}{2}} d t

Commented by Mastermind last updated on 24/Apr/22

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