2xyy-x-1-y-2-x-2-e-x-

Question Number 13649 by ajfour last updated on 22/May/17

2 {x y y}^{'} + (x - 1) y^{2} = x^{2} e^{x}

Answered by ajfour last updated on 22/May/17

l e t y^{2} = t x \Rightarrow 2 {y y}^{'} = t + x \frac{d t}{d x}

s u b s t i t u t i n g

t x + x^{2} \frac{d t}{d x} + {t x}^{2} - t x = x^{2} e^{x}

\frac{d t}{d x} + t = e^{x}; o r x^{2} = 0

\Rightarrow e^{x} \frac{d t}{d x} + {t e}^{x} = e^{2 x}

\int d ({t e}^{x}) = \int e^{2 x} d x

{t e}^{x} = \frac{e^{2 x}}{2} + c

y^{2} = \frac{{x e}^{x}}{2} + {c x e}^{- x}

y = \sqrt{\frac{{x e}^{x}}{2} + {c x e}^{- x}} .

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