If dy/dx = e^(x-y) + x^2e^(-y), then the solution of the differential equation will be:- Option 1: e^y = e^x + x^2/2 + C, where C is an arbitrary constant Option 2: e^y = e^x + x^3/3 + C, where C is an arbitrary constant Option 3: e^y = e^x + 2x + C, where C is an arbitrary constant Option 4: e^y = e^x + 3x^2 + C, where C is an arbitrary constant

Answer: e^y = e^x + x^3/3 + C, where C is an arbitrary constant. Solution: Factor the right side as e^(-y)(e^x+x^2). Separating variables gives e^y dy = (e^x+x^2)dx. Integrating, e^y = e^x + x^3/3 + C.

Calculus

Differential Equations

Easy

applied

If $\dfrac{dy}{dx} = e^{x-y} + x^2e^{-y}$ , then the solution of the differential equation will be:-

$e^y = e^x + \dfrac{x^2}{2} + C$ , where $C$ is an arbitrary constant

$e^y = e^x + \dfrac{x^3}{3} + C$ , where $C$ is an arbitrary constant

$e^y = e^x + 2x + C$ , where $C$ is an arbitrary constant

$e^y = e^x + 3x^2 + C$ , where $C$ is an arbitrary constant

✅ Correct Option: 2

Factor the right side as $e^{-y}(e^x+x^2)$ .

Separating variables gives $e^y dy = (e^x+x^2)dx$ .

Integrating, $e^y = e^x + \dfrac{x^3}{3} + C$ .

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(where $C$ is an arbitrary constant)

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If dydx=ex−y+x2e−y\dfrac{dy}{dx} = e^{x-y} + x^2e^{-y}dxdy​=ex−y+x2e−y, then the solution of the differential equation will be:-

Solution

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