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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 (Commerce) Class 12 - Mathematics

ConceptGeneral and Particular Solutions of a Differential Equation

#### 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,
$y \times I . F . = \int\left( Q \times I . F . \right) dx + C$
$\Rightarrow y e^x = - \int e^x \times \frac{2x}{e^x}dx + C$
$\Rightarrow y e^x = - 2\int x dx + C$
$\Rightarrow y e^x = - x^2 + C$
$\Rightarrow y e^x + x^2 = C$

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Solution 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.
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