solve-the-d-e-y-2xy-sinx-e-x-2-with-initial-condition-y-o-1-

Question Number 27801 by abdo imad last updated on 14/Jan/18

s o l v e t h e d . e . y^{^{'}} - 2 x y = s i n x e^{x^{2}} w i t h i n i t i a l c o n d i t i o n

y (o) = 1 .

Commented by abdo imad last updated on 19/Jan/18

h e \Rightarrow y^{^{'}} - 2 x y = 0 \Leftrightarrow \frac{y^{^{'}}}{y} = 2 x \Leftrightarrow l n / y / = x^{2} + k

\Leftrightarrow y = λ e^{x^{2}} l e t u s e t h e m . v . c m e t h o d

y^{^{'}} = λ^{^{'}} e^{x^{2}} + 2 λ x e^{x^{2}}

a n d (e) \Leftrightarrow λ^{^{'}} e^{x^{2}} + 2 λ x e^{x^{2}} - 2 λ x e^{x^{2}} = s i n x e^{x^{2}}

\Leftrightarrow λ^{^{'}} = s i n x \Leftrightarrow λ = \int s i n x d x + k = - c o s x + k a n d

y (x) = (- c o s x + k) e^{x^{2}} y (0) = 1 \Rightarrow k - 1 = 1 \Rightarrow k = 2

f i n a l l y y (x) = (2 - c o s x) e^{x^{2}} .