sove-the-de-x-2-y-2x-3-y-sin-x-2-with-y-1-2-and-y-1-1-

Question Number 63894 by mathmax by abdo last updated on 10/Jul/19

s o v e t h e (d e) x^{2} y^{^{'}} - (2 x + 3) y = s i n (x^{2}) w i t h y (1) = 2 a n d

y^{^{'}} (1) = 1 .

Commented by mathmax by abdo last updated on 11/Jul/19

(h e) \to x^{2} y^{^{'}} - (2 x + 3) y = 0 \Rightarrow x^{2} y^{^{'}} = (2 x + 3) y \Rightarrow

\frac{y^{^{'}}}{y} = \frac{2 x + 3}{x^{2}} = \frac{2}{x} + \frac{3}{x^{2}} \Rightarrow l n ∣ y ∣= 2 l n ∣ x ∣ - \frac{3}{x} + c \Rightarrow

y = K x^{2} e^{- \frac{3}{x}} l e t u s e m v c m e t h o d

y^{^{'}} = K^{^{'}} x^{2} e^{- \frac{3}{x}} + K {2 x e^{- \frac{3}{x}} + x^{2} \frac{3}{x^{2}} e^{- \frac{3}{x}}}

= K^{^{'}} x^{2} e^{- \frac{3}{x}} + K {2 x + 3} e^{- \frac{3}{x}}

(e) \Rightarrow K^{^{'}} x^{4} e^{- \frac{3}{x}} + {K x}^{2} (2 x + 3) e^{- \frac{3}{x}} - (2 x + 3) {K x}^{2} e^{- \frac{3}{x}} = s i n (x^{2}) \Rightarrow

K^{^{'}} x^{4} e^{- \frac{3}{x}} = s i n (x^{2}) \Rightarrow K^{^{'}} = \frac{s i n (x^{2}) e^{\frac{3}{x}}}{x^{4}} \Rightarrow

K (x) = \int_{1}^{x} \frac{s i n (t^{2}) e^{\frac{3}{t}}}{t^{4}} d t + λ

λ = K (1) w e h a v e y (1) = K (1) e^{- 3} \Rightarrow K (1) = e^{3} y (1) \Rightarrow

K (x) = \int_{1}^{x} \frac{s i n (t^{2}) e^{\frac{3}{t}}}{t^{4}} d t + 2 e^{3} \Rightarrow

y (x) = x^{2} e^{- \frac{3}{x}} {\int_{1}^{x} \frac{e^{\frac{3}{t}} s i n (t^{2})}{t^{4}} d t + 2 e^{3}} \dots .