Form the differential equation corresponding to $y = a x^{2} + b x, (a, b)$ .

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Solution

$\begin{matrix} y = a x^{2} + b x (a, b) (a, b) t w o c o n s tan t s t h e n, r e q u i r e d d i f f e r e n t i a l e q u a t i o n s a r e i n sec o n d o r d e r \frac{y}{x} = a x + b d i f f e r e n t i a t e w . r t o x \Rightarrow \frac{d (y / x)}{d x} = \frac{a d x}{d x} + \frac{d b}{d x} \Rightarrow \frac{x d y / d x - y \frac{d x}{d x}}{x^{2}} = a + 0 \Rightarrow \frac{x d y / d x}{x^{2}} \frac{- y}{x^{2}} = 0 \Rightarrow \frac{d y / d x}{x} - y / x^{2} = a a g a i n d i f f e r e n t i a t e w . r . t o x \Rightarrow x \cdot \frac{\frac{d}{d x} (d y / d x) - d y / d x (\frac{d x}{d x})}{x^{2}} \frac{- ⎡ ⎢ ⎣ x^{2} d y / d x \frac{- y d x^{2}}{d x} ⎤ ⎥ ⎦}{x^{4}} = 0 \frac{x \cdot d^{2} y / d x^{2} - d y / d x}{x^{2}} \frac{- x^{2} d y / d x + y}{x^{4}} = 0 x d^{2} y / d x^{2} - d y / d x \frac{- x^{2} d y / d x}{x^{2}} + \frac{y}{x^{2}} = 0 x d^{2} y / d x^{2} - d y / d x - d y / d x + \frac{y}{x^{2}} = 0 x d^{2} y / d x^{2} - 2 d y / d x + y / x^{2} = 0 r e q u i r e d d i f f e r e n t i a l e q u a t i o n . A n s . \end{matrix}$

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