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# The General Solution of the Differential Equation Ex Dy + (Y Ex + 2x) Dx = 0 is - CBSE (Arts) 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

• x ey + x2 = C

• x ey + y2 = C

• y ex + x2 = C

• y ey + x2 = C

#### Solution

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 Concept: General and Particular Solutions of a Differential Equation.
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