The equation of one of the curves whose slope | vedclass

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(B) The slope of the curve at any point $(x, y)$ is given by $\frac{dy}{dx} = y + 2x$. This is a linear differential equation of the form $\frac{dy}{d

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$y=2(e^x-x-1)$

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(B) The slope of the curve at any point $(x, y)$ is given by $\frac{dy}{dx} = y + 2x$. This is a linear differential equation of the form $\frac{dy}{dx} - y = 2x$. The integrating factor is $IF = e^{\int -1 dx} = e^{-x}$. Multiplying both sides by $e^{-x}$,we get $e^{-x} \frac{dy}{dx} - y e^{-x} = 2x e^{-x}$. This can be written as $\frac{d}{dx}(y e^{-x}) = 2x e^{-x}$. Integrating both sides with respect to $x$: $y e^{-x} = \int 2x e^{-x} dx$. Using integration by parts: $\int 2x e^{-x} dx = 2x(-e^{-x}) - \int 2(-e^{-x}) dx = -2x e^{-x} - 2e^{-x} + C$. So,$y e^{-x} = -2x e^{-x} - 2e^{-x} + C$. Multiplying by $e^x$,we get $y = -2x - 2 + C e^x$. For a curve passing through the origin $(0,0)$,$0 = 0 - 2 + C(1) \Rightarrow C = 2$. Thus,$y = 2e^x - 2x - 2 = 2(e^x - x - 1)$.

The equation of one of the curves whose slope at any point is equal to $y+2x$ is

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