y'' = f(x,y') ........ a

الكاتب Muhanad

$y'' = f(x,y')$ هذه المعادلة محلولة بالنسبة للمشتق الثاني وهي خالية من التالبع . وفكرة حلها هي تحويلها الى الدرجة الأولى وذلك بأن نفرض :

$y' = z\quad\Rightarrow\quad y'' = \frac{{dy'}}{{dx}} = z'$

وبالتالي يكون : $z' = f(x,z)$ وهي معادلة من الدرجة الأولى .

مثال:

$\begin{array}{l}\\ y'' = y'^{^2 } \\ \\ y' = z\quad \Rightarrow \quad y'' = \frac{{dy'}}{{dx}} = z' \\ \\ z' = z^2 \quad \Rightarrow \quad \frac{{z'}}{{z^2 }} = 1 \Rightarrow z = \frac{1}{{c_1 - x}} \\ \\ \Rightarrow \mathop {y'}\limits^{\mathop \uparrow \limits^z } = \frac{1}{{c_1 - x}} \Rightarrow y = \ln \left| {c_1 - x} \right| + c_2 \\ \end{array}$

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Muhanad مشرف في الرياضيات رمز

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