y'' = f(y) ........ b

الكاتب Muhanad

المعادلة $y'' = f(y)$ ..

سنعمل على تخفيض رتبتها من خلال ضرب طرفي المعادلة بـ $2y'$ والمكاملة للطرفين .

$\begin{array}{l}2y'.y'' = f(y) \\ \\ 2y'\frac{{dy'}}{{dx}} = 2f(y).\frac{{dy}}{{dx}}\quad \Rightarrow \quad 2y'.dy' = 2f(y).dy \\ \\ \Rightarrow y'^{^2 } = \int {2f(y).dy} \\ \end{array}$

مثال:

$y'' = 4y$

نضرب الطرفين بـ $2y'$ والمكاملة للطرفين وبالتالي :

$\begin{array}{l}2y'y'' = 8yy' \\ \\ y'^{^2 } = 4y^2 \Rightarrow y' = \mp 2y + c_1 \\ \\ \frac{{y'}}{{c_1 \mp 2y}} = 1 \Rightarrow \ln \left| {c_1 \mp 2y} \right| = \mp 2x + c_2 \\ \end{array}$

نبذة عن كاتب الموضوع

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Muhanad مشرف في الرياضيات رمز

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