y-y-sec-x-

Question Number 153542 by EDWIN88 last updated on 08/Sep/21

y^{‴} + y^{'} = \sec x

Answered by puissant last updated on 08/Sep/21

y^{‴} + y^{'} = s e c x

\Rightarrow y^{″} + y = l n (t a n (\frac{x}{2} + \frac{π}{2})) + C_{1}

\Rightarrow (D^{2} + D) y = l n (t a n (\frac{x}{2} + \frac{π}{2})) + C_{1}

\Rightarrow (D + 1) D y = l n (t a n (\frac{x}{2} + \frac{π}{2})) + C_{1}

\Rightarrow (D + 1) y = \int l n (t a n (\frac{x}{2} + \frac{π}{2})) d x + C_{1} x + C_{2}

\Rightarrow e^{x} (D + 1) y = e^{x} \int l n (t a n (\frac{x}{2} + \frac{π}{2})) d x + C_{1} {x e}^{x} + C_{2} e^{x}

\Rightarrow D (e^{x} y) = e^{x} \int l n (t a n (\frac{x}{2} + \frac{π}{2})) d x + C_{1} {x e}^{x} + C_{2} e^{x}

\Rightarrow e^{x} y = \int e^{x} \int l n (t a n (\frac{x}{2} + \frac{π}{2})) d x d x + \frac{C_{1} x^{2} e^{x}}{2} + C_{2} {x e}^{x} + C_{3}

\Rightarrow y = e^{- x} \int e^{x} \int l n (t a n (\frac{x}{2} + \frac{π}{2})) d x d x + \frac{C_{1} x^{2}}{2} + C_{2} x + C_{3} e^{- x}

\dots \dots \dots \dots \dots \dots

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