For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
$y = a e^{x} + b e^{- x} + x^{2}$ : $x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0$

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Solution

$G i v e n e q u a t i o n$
$y = a e^{x} + b e^{- x} + x^{2}$
$D i f f e r e n t i a t i n g w r t t o x$
$\frac{d y}{d x} = a e^{x} - b e^{- x} + 2 x . . . . . . . . . . . . . . . (1)$
$D i f f e r e n t i a t i n g a g a i n w r t x$
$\frac{d^{2} y}{d x^{2}} = a e^{x} + b e^{- x} + 2.................. (2)$
$G i v e n D i f f e r e n t i a l e q u a t i o n$
$x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0$
$T o c h e c k w h e t h e r g i v e n e q u a t i o n s a t i s f i e s t h e D E s u b s t i t u t e v a l u e s o f y, \frac{d y}{d x}, \frac{d^{2} y}{d x^{2}} i n D E$
$x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2$
$= x (a e^{x} + b e^{- x} + 2) + 2 (a e^{x} - b e^{- x} + 2 x) - x (a e^{x} + b e^{- x} + x^{2}) + x^{2} - 2$
$C a n c e l i n g t e r m s w e w e g e t$
$6 x + 2 a e^{x} - 2 b e^{- x} - x^{3} + x^{2} - 2 \neq 0$
$H e n c e g i v e n f u n c t i o n n o t s o l u t i o n o f D E$

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Similar questions

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i)

(ii)

(iii)

(iv)

For the question given below verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

$y = a e^{x} + b e^{- x} + x^{2}; x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0$

Q. Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(i)

y = e^{x} (a cos x + b s i n x)

\frac{d^{2} y}{d x^{2}} - 2 \frac{d y}{d x} + 2 y = 0

(ii)

y = x sin 3 x

\frac{d^{2} y}{d x^{2}} + 9 y - 6 cos 3 x = 0

(iii)

x^{2} = 2 y^{2} log y

(x^{2} + y^{2}) \frac{d y}{d x} - x y = 0

For the following question verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

$y = e^{x} (a c o s x + b s i n x) : \frac{d^{2} y}{d x^{2}} - 2 \frac{d y}{d x} + 2 y = 0$

Q. For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation. (i) (ii) (iii) (iv)

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For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.y=aex+be−x+x2 : xd2ydx2+2dydx−xy+x2−2=0

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
$y = a e^{x} + b e^{- x} + x^{2}$ : $x \frac{d^{2} y}{d x^{2}} + 2 \frac{d y}{d x} - x y + x^{2} - 2 = 0$