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Solution for The General Solution of the Differential Equation Ex Dy + (Y Ex + 2x) Dx = 0 is (A) X Ey + X2 = C (B) X Ey + Y2 = C (C) Y Ex + X2 = C (D) Y Ey + X2 = C - CBSE (Science) Class 12 - Mathematics

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Question

The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is
(a) x ey + x2 = C
(b) x ey + y2 = C
(c) y ex + x2 = C
(d) y ey + x2 = C

Solution

(c) y ex + x2 = C
We have,
ex dy + (yex + 2x) dx = 0
\[\text{ Dividing both sides by }e^x dx, \text{ we get }\]
\[\frac{dy}{dx} + \left( y + \frac{2x}{e^x} \right) = 0\]
\[ \Rightarrow \frac{dy}{dx} + y = - \frac{2x}{e^x}\]
\[\text{ Comparing with }\frac{dy}{dx} + Py = Q,\text{ we get }\]
\[P = 1\]
\[Q = - \frac{2x}{e^x}\]
Now, 
\[I . F . = e^\int dx = e^x \]
Solution is given by, 
\[y \times I . F . = \int\left( Q \times I . F . \right) dx + C\]
\[ \Rightarrow y e^x = - \int e^x \times \frac{2x}{e^x}dx + C\]
\[ \Rightarrow y e^x = - 2\int x dx + C\]
\[ \Rightarrow y e^x = - x^2 + C\]
\[ \Rightarrow y e^x + x^2 = C \]

  Is there an error in this question or solution?
Solution for question: The General Solution of the Differential Equation Ex Dy + (Y Ex + 2x) Dx = 0 is (A) X Ey + X2 = C (B) X Ey + Y2 = C (C) Y Ex + X2 = C (D) Y Ey + X2 = C concept: General and Particular Solutions of a Differential Equation. For the courses CBSE (Science), CBSE (Arts), CBSE (Commerce), PUC Karnataka Science
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