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New answer posted

3 months ago

Differential Equations Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Solve: $x \frac{d y}{d x} = y (l o g y - l o g x + 1) .$

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alok kumar singh

Contributor-Level 10

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t x \frac{d y}{d x} = y (l o g y - l o g x + 1) \\ \Rightarrow x \frac{d y}{d x} = y [l o g (\frac{y}{x}) + 1] \\ \Rightarrow \frac{d y}{d x} = \frac{y}{x} [l o g (\frac{y}{x}) + 1] \\ Since, i t i s a d i f f e r e n t i a l e q u a t i o n \\ S o, p u t y = v x \Rightarrow \frac{d y}{d x} = v + x . \frac{d v}{d x} \\ \Rightarrow v + x . \frac{d v}{d x} = \frac{v x}{x} [l o g (\frac{v x}{x}) + 1] \\ \Rightarrow v + x . \frac{d v}{d x} = v [l o g v + 1] \\ \Rightarrow x . \frac{d v}{d x} = v [l o g v + 1] - v \Rightarrow x . \frac{d v}{d x} = v [l o g v + 1 - 1] \\ \Rightarrow x . \frac{d v}{d x} = v . l o g v \Rightarrow \frac{d v}{v l o g v} = \frac{d x}{x} \\ I n t e g r a t i n g b o t h s i d e s, w e h a v e \\ \int \frac{d v}{v l o g v} = \int \frac{d x}{x} \\ P u t l o g v = t o n L . H . S . \\ \Rightarrow \frac{1}{v} d v = d t \\ ∴ \int \frac{d t}{t} = \int \frac{d x}{x} \\ \Rightarrow l o g | t | = l o g | x | + l o g c \\ \Rightarrow l o g | l o g v | = l o g x c \Rightarrow l o g v = x c \\ \Rightarrow l o g (\frac{y}{x}) = x c \\ H e n c e, t h e r e q u i r e d s o l u t i o n i s l o g (\frac{y}{x}) = x c . \end{array}$

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3 months ago

Differential Equations Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Find the equation of a curve passing through the point $(1, 1)$ . If the tangent drawn at any point $P (x, y)$ on the curve meets the coordinate axes at $A$ and $B$ such that $P$ is the midpoint of $A B$ .

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alok kumar singh

Contributor-Level 10

This is a Long Answer Type Questions as classified in NCERT Exemplar

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New answer posted

3 months ago

Differential Equations Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Find the equation of a curve passing through the origin if the slope of the tangent to the curve at any point $(x, y)$ is equal to the square of the difference of the abscissa and ordinate of the point.

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alok kumar singh

Contributor-Level 10

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} H e r e, s l o p e o f t h e o f t h e c u r v e = \frac{d y}{d x} \\ a n d t h e d i f f e r e n c e b e t w e e n t h e a b s c i s s a a n d o r d i n a t e = x - y . \\ ∴ A s p e r t h e c o n d i t i o n, \frac{d y}{d x} = {(x - y)}^{2} \\ P u t x - y = v \\ 1 - \frac{d y}{d x} = \frac{d v}{d x} \\ ∴ \frac{d y}{d x} = 1 - \frac{d v}{d x} \\ ∴ t h e e q u a t i o n b e c o m e s \\ 1 - \frac{d v}{d x} = v^{2} \Rightarrow \frac{d v}{d x} = 1 - v^{2} \Rightarrow \frac{d v}{1 - v^{2}} = d x \\ I n t e g r a t i n g b o t h s i d e s, w e g e t \\ \int \frac{d v}{1 - v^{2}} = \int d x \\ \Rightarrow \frac{1}{2} l o g | \frac{1 + v}{1 - v} | = x + c \Rightarrow \frac{1}{2} l o g | \frac{1 + x - y}{1 - x + y} | = x + c \dots (1) \\ Since, t h e c u r v e i s t h r o u g h (0, 0), t h e n \\ \Rightarrow \frac{1}{2} l o g | \frac{1 + 0 - 0}{1 - 0 + 0} | = 0 + c \Rightarrow c = 0 \\ ∴ O n p u t t i n g c = 0 i n e q . (1) w e g e t \\ \frac{1}{2} l o g | \frac{1 + x - y}{1 - x + y} | = x \Rightarrow l o g | \frac{1 + x - y}{1 - x + y} | = 2 x \\ ∴ \frac{1 + x - y}{1 - x + y} = e^{2 x} \\ \Rightarrow (1 + x - y) = e^{2 x} (1 - x + y) \\ H e n c e, t h e r e q u i r e d e q u a t i o n i s (1 + x - y) = e^{2 x} (1 - x + y) . \end{array}$

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3 months ago

Differential Equations Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Find the equation of the curve through the point $(1, 0)$ if the slope of the tangent to the curve at any point $(x, y)$ is $\frac{y - 1}{x (x + 2)} .$

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alok kumar singh

Contributor-Level 10

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t t h e s l o p e o f tangent t o a c u r v e a t (x, y) i s \\ \frac{d y}{d x} = \frac{y - 1}{x^{2} + x} \Rightarrow \frac{d y}{y - 1} = \frac{d x}{x^{2} + x} \\ I n t e g r a t i n g b o t h s i d e s, w e h a v e \\ \int \frac{d y}{y - 1} = \int \frac{d x}{x^{2} + x} \\ \Rightarrow \int \frac{d y}{y - 1} = \int \frac{d x}{x^{2} + x + \frac{1}{4} - \frac{1}{4}} [M a k i n g p e r f e c t s q u a r e] \\ \Rightarrow \int \frac{d y}{y - 1} = \int \frac{d x}{{(x + \frac{1}{2})}^{2} - {(\frac{1}{2})}^{2}} \\ \Rightarrow l o g | y - 1 | = \frac{1}{2 * \frac{1}{2}} l o g | \frac{x + \frac{1}{2} - \frac{1}{2}}{x + \frac{1}{2} + \frac{1}{2}} | + l o g c \\ \Rightarrow l o g | y - 1 | = l o g | \frac{x}{x + 1} | + l o g c \\ \Rightarrow l o g | y - 1 | = l o g | c (\frac{x}{x + 1}) | \\ ∴ y - 1 = \frac{c x}{x + 1} \Rightarrow (y - 1) (x + 1) = c x \\ Since, t h e l i n e i s passing t h r o u g h t h e point (1, 0), t h e n \\ (0 - 1) (1 + 1) = c (1) \Rightarrow c = 2 \\ H e n c e, t h e r e q u i r e d s o l u t i o n i s (y - 1) (x + 1) = 2 x . \end{array}$

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New answer posted

3 months ago

Differential Equations Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Find the equation of a curve passing through $(2, 1)$ if the slope of the tangent to the curve at any point $(x, y)$ is $\frac{x^{2} + y^{2}}{2 x y} .$

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alok kumar singh

Contributor-Level 10

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t t h e s l o p e o f tangent t o a c u r v e a t (x, y) i s \\ \frac{d y}{d x} = \frac{x^{2} + y^{2}}{2 x y} \\ I t i s a homogeneous d i f f e r e n t i a l e q u a t i o n \\ S o, p u t y = v x \Rightarrow \frac{d y}{d x} = v + x . \frac{d v}{d x} \\ \Rightarrow v + x . \frac{d v}{d x} = \frac{x^{2} + v^{2} x^{2}}{2 x . v x} \\ \Rightarrow v + x . \frac{d v}{d x} = \frac{1 + v^{2}}{2 v} \\ \Rightarrow x . \frac{d v}{d x} = \frac{1 + v^{2}}{2 v} - v \Rightarrow x . \frac{d v}{d x} = \frac{1 + v^{2} - 2 v^{2}}{2 v} \\ \Rightarrow x . \frac{d v}{d x} = \frac{1 - v^{2}}{2 v} \Rightarrow \frac{2 v}{1 - v^{2}} d v = \frac{d x}{x} \\ I n t e g r a t i n g b o t h s i d e s, w e h a v e \\ \int \frac{2 v}{1 - v^{2}} d v = \int \frac{d x}{x} \\ \Rightarrow - l o g | 1 - v^{2} | = l o g x + l o g c \\ \Rightarrow - l o g | 1 - \frac{y^{2}}{x^{2}} | = l o g x + l o g c \Rightarrow - l o g | \frac{x^{2} - y^{2}}{x^{2}} | = l o g x + l o g c \\ \Rightarrow l o g | \frac{x^{2}}{x^{2} - y^{2}} | = l o g | x c | \Rightarrow \frac{x^{2}}{x^{2} - y^{2}} = x c \\ Since, t h e c u r v e i s passing t h r o u g h t h e point (2, 1) \\ ∴ \frac{{(2)}^{2}}{{(2)}^{2} - {(1)}^{2}} = 2 c \Rightarrow \frac{4}{3} = 2 c \Rightarrow c = \frac{2}{3} \\ H e n c e, t h e r e q u i r e d e q u a t i o n i s \frac{x^{2}}{x^{2} - y^{2}} = \frac{2}{3} x \Rightarrow 2 (x^{2} - y^{2}) = 3 x \end{array}$

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New answer posted

3 months ago

Differential Equations Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Find the general solution of $\frac{d y}{d x} - 3 y = s i n 2 x$

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alok kumar singh

Contributor-Level 10

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n e q u a t i o n i s \frac{d y}{d x} - 3 y = s i n 2 x \\ H e r e, P = - 3 a n d Q = s i n 2 x \\ I n t e g r a t i n g f a c t o r I . F = e^{\int P d x} = e^{\int - 3 . d x} = e^{- 3 x} \\ ∴ S o l u t i o n o f t h e e q u a t i o n i s \\ y * I . F . = \int Q * I . F . d x + c \\ \Rightarrow y . e^{- 3 x} = \int s i n 2 x . e^{- 3 x} d x + c \\ L e t I = \int \underset{I}{s i n 2 x} . \underset{I I}{e^{- 3 x}} d x \\ \Rightarrow I = s i n 2 x . \int e^{- 3 x} d x - \int (D (s i n 2 x) . \int e^{- 3 x} d x) d x \\ \Rightarrow I = s i n 2 x . \frac{e^{- 3 x}}{- 3} - \int 2 c o s 2 x . \frac{e^{- 3 x}}{- 3} d x \\ \Rightarrow I = \frac{e^{- 3 x}}{- 3} s i n 2 x + \frac{2}{3} \int \underset{I}{c o s 2 x} . \underset{I I}{e^{- 3 x}} d x \\ \Rightarrow I = \frac{e^{- 3 x}}{- 3} s i n 2 x + \frac{2}{3} [c o s 2 x . \int e^{- 3 x} d x - \int (D (c o s 2 x) . \int e^{- 3 x} d x) d x] \\ \Rightarrow I = \frac{e^{- 3 x}}{- 3} s i n 2 x + \frac{2}{3} [c o s 2 x . \frac{e^{- 3 x}}{- 3} - \int 2 s i n 2 x . \frac{e^{- 3 x}}{- 3} d x] \\ \Rightarrow I = \frac{e^{- 3 x}}{- 3} s i n 2 x - \frac{2}{9} c o s 2 x . e^{- 3 x} - \frac{4}{9} \int s i n 2 x . e^{- 3 x} d x \\ \Rightarrow I = \frac{e^{- 3 x}}{- 3} s i n 2 x - \frac{2}{9} c o s 2 x . e^{- 3 x} - \frac{4}{9} I \\ \Rightarrow I + \frac{4}{9} I = \frac{e^{- 3 x}}{- 3} s i n 2 x - \frac{2}{9} e^{- 3 x} c o s 2 x \\ \Rightarrow \frac{13}{9} I = - \frac{1}{9} [3 e^{- 3 x} s i n 2 x + 2 e^{- 3 x} c o s 2 x] \\ \Rightarrow I = - \frac{1}{13} e^{- 3 x} [3 s i n 2 x + 2 c o s 2 x] \\ ∴ T h e e q u a t i o n b e c o m e s \\ \Rightarrow y . e^{- 3 x} = - \frac{1}{13} e^{- 3 x} [3 s i n 2 x + 2 c o s 2 x] + c \\ ∴ y = - \frac{1}{13} [3 s i n 2 x + 2 c o s 2 x] + c . e^{3 x} \\ H e n c e, t h e r e q u i r e d s o l u t i o n i s y = - [\frac{3 s i n 2 x + 2 c o s 2 x}{13}] + c . e^{3 x} \end{array}$

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New answer posted

3 months ago

Differential Equations Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Solve: $\frac{d y}{d x} = c o s (x + y) + s i n (x + y)$ . [Hint: Substitute $x + y = z$ ]

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alok kumar singh

Contributor-Level 10

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t \frac{d y}{d x} = c o s (x + y) + s i n (x + y) \\ P u t x + y = v, o n d i f f e r e n t i a t i n g w . r . t . x, w e g e t, \\ 1 + \frac{d y}{d x} = \frac{d v}{d x} \\ ∴ \frac{d y}{d x} = \frac{d v}{d x} - 1 \\ ∴ \frac{d v}{d x} - 1 = c o s v + s i n v \\ \Rightarrow \frac{d v}{d x} = c o s v + s i n v + 1 \\ \Rightarrow \frac{d v}{c o s v + s i n v + 1} = d x \\ I n t e g r a t i n g b o t h s i d e s, w e h a v e \\ \int \frac{d v}{c o s v + s i n v + 1} = \int 1 . d x \\ \Rightarrow \int \frac{d v}{(\frac{1 - t a n^{2} \frac{v}{2}}{1 + t a n^{2} \frac{v}{2}} + \frac{2 t a n \frac{v}{2}}{1 + t a n^{2} \frac{v}{2}} + 1)} = \int 1 . d x \\ \Rightarrow \int \frac{1 + t a n^{2} \frac{v}{2}}{(1 - t a n^{2} \frac{v}{2} + 2 t a n \frac{v}{2} + 1)} d v = \int 1 . d x \\ \Rightarrow \int \frac{s e c^{2} \frac{v}{2}}{2 + 2 t a n \frac{v}{2}} d v = \int 1 . d x \\ P u t 2 + 2 t a n \frac{v}{2} = t \\ 2 . \frac{1}{2} s e c^{2} \frac{v}{2} d v = d t \Rightarrow s e c^{2} \frac{v}{2} d v = d t \\ \Rightarrow \int \frac{d t}{2} = \int 1 . d x \\ \Rightarrow l o g | t | = x + c \\ \Rightarrow l o g | 2 + 2 t a n \frac{v}{2} | = x + c \\ \Rightarrow l o g | 2 + 2 t a n (\frac{x + y}{2}) | = x + c \\ \Rightarrow l o g 2 [1 + t a n (\frac{x + y}{2})] = x + c \\ \Rightarrow l o g 2 + l o g [1 + t a n (\frac{x + y}{2})] = x + c \\ \Rightarrow l o g [1 + t a n (\frac{x + y}{2})] = x + c - l o g 2 \\ H e n c e, t h e r e q u i r e d s o l u t i o n i s l o g [1 + t a n (\frac{x + y}{2})] = x + K [c - l o g 2 = K] \end{array}$

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New answer posted

3 months ago

Differential Equations Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Find the general solution of $(1 + t a n y) (d x- d y) + 2 x d y = 0$ .

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alok kumar singh

Contributor-Level 10

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n t h a t (1 + t a n y) (d x - d y) + 2 x d y = 0 \\ \Rightarrow (1 + t a n y) d x - (1 + t a n y) d y + 2 x d y = 0 \\ \Rightarrow (1 + t a n y) d x - (1 + t a n y - 2 x) d y = 0 \\ \Rightarrow (1 + t a n y) \frac{d x}{d y} = (1 + t a n y - 2 x) \Rightarrow \frac{d x}{d y} = \frac{1 + t a n y - 2 x}{1 + t a n y} \\ \Rightarrow \frac{d x}{d y} = 1 - \frac{2 x}{1 + t a n y} \Rightarrow \frac{d x}{d y} + \frac{2 x}{1 + t a n y} = 1 \\ H e r e, P = \frac{2}{1 + t a n y} a n d Q = 1 \\ I n t e g r a t i n g f a c t o r I . F = e^{\int P d y} = e^{\int \frac{2}{1 + t a n y} . d y} = e^{\int \frac{2 c o s y}{s i n y + c o s y} . d y} \\ = e^{\int \frac{s i n y + c o s y - s i n y + c o s y}{s i n y + c o s y} . d y} = e^{\int (1 + \frac{c o s y - s i n y}{s i n y + c o s y}) . d y} \\ = e^{\int 1 . d y} . e^{\int \frac{c o s y - s i n y}{s i n y + c o s y} . d y} = e^{y} . e^{l o g (s i n y + c o s y)} = e^{y} . (s i n y + c o s y) \\ ∴ S o l u t i o n o f t h e e q u a t i o n i s \\ x * I . F . = \int Q * I . F . d y + c \\ \Rightarrow x . e^{y} . (s i n y + c o s y) = \int 1 . e^{y} . (s i n y + c o s y) d y + c \\ \Rightarrow x . e^{y} . (s i n y + c o s y) = e^{y} . s i n y + c [? \int e^{x} [f (x) + f^{'} (x)] d x = e^{x} f (x) + c] \\ \Rightarrow x (s i n y + c o s y) = s i n y + c . e^{- y} \\ H e n c e, t h e r e q u i r e d s o l u t i o n i s x (s i n y + c o s y) = s i n y + c . e^{- y} \end{array}$

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New answer posted

3 months ago

Differential Equations Maths NCERT Exemplar Solutions Class 12th Chapter Nine

Solve: $(\frac{d y}{d x}) + x y = x (s i n x + l o g x)$ .

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alok kumar singh

Contributor-Level 10

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} T h e g i v e n d i f f e r e n t i a l e q u a t i o n i s y + \frac{d}{d x} (x y) = x (s i n x + l o g x) \\ \Rightarrow y + x . \frac{d y}{d x} + y = x (s i n x + l o g x) \\ \Rightarrow x \frac{d y}{d x} = x (s i n x + l o g x) - 2 y \\ \Rightarrow \frac{d y}{d x} = (s i n x + l o g x) - \frac{2 y}{x} \\ \Rightarrow \frac{d y}{d x} + \frac{2}{x} y = (s i n x + l o g x) \\ H e r e, P = \frac{2}{x} a n d Q = (s i n x + l o g x) \\ I n t e g r a t i n g f a c t o r I . F = e^{\int P d x} = e^{\int \frac{2}{x} . d x} = e^{2 l o g x} = e^{l o g x^{2}} = x^{2} \\ ∴ S o l u t i o n o f t h e e q u a t i o n i s \\ y * I . F . = \int Q * I . F . d x + c \\ \Rightarrow y . x^{2} = \int (s i n x + l o g x) x^{2} d x + c \dots (1) \\ L e t I = \int (s i n x + l o g x) x^{2} d x \\ = \int \underset{I}{x^{2}} \underset{I I}{s i n x} d x + \int \underset{I I}{x^{2}} \underset{I}{l o g x} d x \\ = [x^{2} . \int s i n x d x - \int (D (x^{2}) . \int s i n x d x) d x] + [l o g x . \int x^{2} d x - \int (D (l o g x) . \int x^{2} d x) d x] \\ = [x^{2} (- c o s x) - 2 \int - x c o s x d x] + [l o g x . \frac{x^{3}}{3} - \int \frac{1}{x} . \frac{x^{3}}{3} d x] \\ = [- x^{2} c o s x + 2 (x s i n x - \int 1 . s i n x d x)] + [\frac{x^{3}}{3} l o g x - \frac{1}{3} \int x^{2} d x] \\ = - x^{2} c o s x + 2 x s i n x + 2 c o s x + \frac{x^{3}}{3} l o g x - \frac{1}{9} x^{3} \\ N o w f r o m e q (1) w e g e t, \\ y . x^{2} = - x^{2} c o s x + 2 x s i n x + 2 c o s x + \frac{x^{3}}{3} l o g x - \frac{1}{9} x^{3} + c \\ ∴ y = - c o s x + \frac{2 s i n x}{x} + \frac{2 c o s x}{x^{2}} + \frac{x l o g x}{3} - \frac{1}{9} x + c . x^{- 2} \\ H e n c e, t h e r e q u i r e d s o l u t i o n i s \\ y = - c o s x + \frac{2 s i n x}{x} + \frac{2 c o s x}{x^{2}} + \frac{x l o g x}{3} - \frac{1}{9} x + c . x^{- 2} \end{array}$

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New answer posted

4 months ago

Differential Equations Class 12th

113. The general solution of the differential equation $e^{x} d y + (y e^{x} + 2 x) d x = 0$ is:

$\begin{array}{l} (A) x e^{y} + x^{2} = C \\ (B) x e^{y} + y^{2} = C \\ (C) y e^{x} + x^{2} = C \\ (D) y e^{y} + x^{2} = C \end{array}$

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Vishal Baghel

Contributor-Level 10

The given differential equation is:

$\begin{array}{l} e^{x} d y + (y e^{x} + 2 x) d x = 0 \\ \Rightarrow e^{x} \frac{d y}{d x} + y e^{x} + 2 x = 0 \\ \Rightarrow \frac{d y}{d x} + y = - 2 x e^{- x} \end{array}$

This is a linear differential equation of the form

$\begin{array}{l} \frac{d y}{d x} + P y = Q, w h e r e P = 1 & Q = - 2 x e^{- x} \\ N o w, I . F . = e^{\int P d x} = e^{\int d x} = e^{x} \end{array}$

The general solution of the given differential equation is given by,

$\begin{array}{l} y (I . F .) = \int (Q * I . F .) d x + C \\ \Rightarrow y e^{x} = \int (- 2 x e^{- x} . e^{x}) d x + C \\ \Rightarrow y e^{x} = - \int 2 x d x + C \\ \Rightarrow y e^{x} = - x^{2} + C \\ \Rightarrow y e^{x} + x^{2} = C \end{array}$

Therefore, option (c) is correct.

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